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/home/sep/Documents/Repositories/pyxu/doc/fair/plugins/pyxu-eigh.rst:164: ERROR: Content block expected for the "raw" directive; none found. /home/sep/Documents/Repositories/pyxu/doc/fair/plugins/pyxu-gradient-descent.rst:164: ERROR: Content block expected for the "raw" directive; none found. -/home/sep/Documents/Repositories/pyxu/doc/guide/funcs.ipynb:129: WARNING: Title underline too short. +/home/sep/Documents/Repositories/pyxu/doc/guide/funcs.ipynb:137: WARNING: Title underline too short. Implicit Functionals: The Undercover Agents 🕵️‍♀️ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ -/home/sep/Documents/Repositories/pyxu/doc/guide/funcs.ipynb:129: WARNING: Title underline too short. +/home/sep/Documents/Repositories/pyxu/doc/guide/funcs.ipynb:137: WARNING: Title underline too short. Implicit Functionals: The Undercover Agents 🕵️‍♀️ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ -/home/sep/Documents/Repositories/pyxu/doc/guide/funcs.ipynb:27: WARNING: File not found: 'api/operator.interop.html#general' +/home/sep/Documents/Repositories/pyxu/doc/guide/funcs.ipynb:27: WARNING: File not found: 'api/operator.interop.html#pyxu.operator.interop.from_source' /home/sep/Documents/Repositories/pyxu/doc/guide/funcs.ipynb:27: WARNING: File not found: 'api/abc.html#pyxu.abc.Map.apply' /home/sep/Documents/Repositories/pyxu/doc/guide/funcs.ipynb:27: WARNING: File not found: 'api/abc.html#pyxu.abc.DiffFunc.grad' /home/sep/Documents/Repositories/pyxu/doc/guide/funcs.ipynb:27: WARNING: File not found: 'api/abc.html#pyxu.abc.ProxFunc.prox' +/home/sep/Documents/Repositories/pyxu/doc/guide/funcs.ipynb:29: WARNING: File not found: 'api/index.html' /home/sep/Documents/Repositories/pyxu/doc/guide/funcs.ipynb:34: WARNING: File not found: 'api/abc.html#pyxu.abc.Func' -/home/sep/Documents/Repositories/pyxu/doc/guide/funcs.ipynb:61: WARNING: File not found: 'api/abc.html#pyxu.abc.DiffFunc' -/home/sep/Documents/Repositories/pyxu/doc/guide/funcs.ipynb:76: WARNING: File not found: 'api/operator.interop.html#pyxu.operator.interop.from_torch' -/home/sep/Documents/Repositories/pyxu/doc/guide/funcs.ipynb:76: WARNING: File not found: 'api/operator.interop.html#pyxu.operator.interop.from_jax' -/home/sep/Documents/Repositories/pyxu/doc/guide/funcs.ipynb:81: WARNING: File not found: 'api/abc.html#pyxu.abc.ProxFunc' -/home/sep/Documents/Repositories/pyxu/doc/guide/funcs.ipynb:97: WARNING: File not found: 'api/abc.html#pyxu.abc.ProxFunc.moreau_envelope' -/home/sep/Documents/Repositories/pyxu/doc/guide/funcs.ipynb:124: WARNING: File not found: 'api/abc.html#pyxu.abc.ProxDiffFunc' -/home/sep/Documents/Repositories/pyxu/doc/guide/funcs.ipynb:124: WARNING: File not found: 'api/abc.html#pyxu.abc.LinFunc' -/home/sep/Documents/Repositories/pyxu/doc/guide/funcs.ipynb:124: WARNING: File not found: 'api/abc.html#pyxu.abc.QuadraticFunc' +/home/sep/Documents/Repositories/pyxu/doc/guide/funcs.ipynb:34: WARNING: File not found: 'api/abc.html#pyxu.abc.Map.apply' +/home/sep/Documents/Repositories/pyxu/doc/guide/funcs.ipynb:50: WARNING: File not found: 'api/abc.html#pyxu.abc.Func.asloss' +/home/sep/Documents/Repositories/pyxu/doc/guide/funcs.ipynb:63: WARNING: File not found: 'api/abc.html#pyxu.abc.DiffFunc' +/home/sep/Documents/Repositories/pyxu/doc/guide/funcs.ipynb:63: WARNING: File not found: 'api/abc.html#pyxu.abc.DiffFunc.grad' +/home/sep/Documents/Repositories/pyxu/doc/guide/funcs.ipynb:81: WARNING: File not found: 'api/operator.interop.html#pyxu.operator.interop.from_torch' +/home/sep/Documents/Repositories/pyxu/doc/guide/funcs.ipynb:81: WARNING: File not found: 'api/operator.interop.html#pyxu.operator.interop.from_jax' +/home/sep/Documents/Repositories/pyxu/doc/guide/funcs.ipynb:86: WARNING: File not found: 'api/abc.html#pyxu.abc.ProxFunc' +/home/sep/Documents/Repositories/pyxu/doc/guide/funcs.ipynb:86: WARNING: File not found: 'api/abc.html#pyxu.abc.ProxFunc.prox' +/home/sep/Documents/Repositories/pyxu/doc/guide/funcs.ipynb:105: WARNING: File not found: 'api/abc.html#pyxu.abc.ProxFunc.moreau_envelope' +/home/sep/Documents/Repositories/pyxu/doc/guide/funcs.ipynb:132: WARNING: File not found: 'api/abc.html#pyxu.abc.ProxDiffFunc' +/home/sep/Documents/Repositories/pyxu/doc/guide/funcs.ipynb:132: WARNING: File not found: 'api/abc.html#pyxu.abc.LinFunc' +/home/sep/Documents/Repositories/pyxu/doc/guide/funcs.ipynb:132: WARNING: File not found: 'api/abc.html#pyxu.abc.QuadraticFunc' /home/sep/Documents/Repositories/pyxu/doc/guide/fwdops.ipynb:11: WARNING: File not found: 'api/abc.html#pyxu.abc.LinOp' /home/sep/Documents/Repositories/pyxu/doc/guide/fwdops.ipynb:17: WARNING: File not found: 'api/operator.interop.html#general' /home/sep/Documents/Repositories/pyxu/doc/guide/fwdops.ipynb:17: WARNING: File not found: 'api/abc.html#pyxu.abc.Map.apply' /home/sep/Documents/Repositories/pyxu/doc/guide/fwdops.ipynb:17: WARNING: File not found: 'api/abc.html#pyxu.abc.LinOp.adjoint' /home/sep/Documents/Repositories/pyxu/doc/guide/fwdops.ipynb:17: WARNING: File not found: 'api/abc.html#pyxu.abc.DiffMap.jacobian' +/home/sep/Documents/Repositories/pyxu/doc/guide/fwdops.ipynb:19: WARNING: File not found: 'api/index.html' /home/sep/Documents/Repositories/pyxu/doc/guide/fwdops.ipynb:46: WARNING: File not found: 'api/abc.html#pyxu.abc.LinOp' +/home/sep/Documents/Repositories/pyxu/doc/guide/fwdops.ipynb:46: WARNING: File not found: 'api/abc.html#pyxu.abc.Map.apply' +/home/sep/Documents/Repositories/pyxu/doc/guide/fwdops.ipynb:46: WARNING: File not found: 'api/abc.html#pyxu.abc.LinOp.adjoint' /home/sep/Documents/Repositories/pyxu/doc/guide/fwdops.ipynb:55: WARNING: File not found: 'api/abc.html#pyxu.abc.Map.apply' /home/sep/Documents/Repositories/pyxu/doc/guide/fwdops.ipynb:66: WARNING: File not found: 'api/abc.html#pyxu.abc.LinOp.adjoint' /home/sep/Documents/Repositories/pyxu/doc/guide/fwdops.ipynb:100: WARNING: File not found: 'api/abc.html#pyxu.abc.LinOp' @@ -52,46 +60,13 @@ Implicit Functionals: The Undercover Agents 🕵️‍♀️ /home/sep/Documents/Repositories/pyxu/doc/guide/fwdops.ipynb:148: WARNING: File not found: 'api/abc.html#pyxu.abc.DiffMap' /home/sep/Documents/Repositories/pyxu/doc/guide/fwdops.ipynb:153: WARNING: File not found: 'api/abc.html#pyxu.abc.LinOp' /home/sep/Documents/Repositories/pyxu/doc/guide/fwdops.ipynb:153: WARNING: File not found: 'api/abc.html#pyxu.abc.DiffMap' -/home/sep/Documents/Repositories/pyxu/doc/guide/hpc.ipynb:219: WARNING: File not found: 'api/runtime.html#pyxu.runtime.Precision' +/home/sep/Documents/Repositories/pyxu/doc/guide/hpc.ipynb:212: WARNING: File not found: 'api/runtime.html#pyxu.runtime.Precision' /home/sep/Documents/Repositories/pyxu/doc/guide/lipschitz.ipynb:30: WARNING: File not found: 'api/abc.html#pyxu.abc.Map.lipschitz' /home/sep/Documents/Repositories/pyxu/doc/guide/lipschitz.ipynb:30: WARNING: File not found: 'api/abc.html#pyxu.abc.DiffMap.diff_lipschitz' /home/sep/Documents/Repositories/pyxu/doc/guide/lipschitz.ipynb:40: WARNING: File not found: 'api/abc.html#pyxu.abc.Map.estimate_lipschitz' -/home/sep/Documents/Repositories/pyxu/doc/guide/lipschitz.ipynb:298: WARNING: File not found: 'api/index.html' -/home/sep/Documents/Repositories/pyxu/doc/guide/lipschitz.ipynb:298: WARNING: File not found: 'api/abc/arithmetic.html#' -/home/sep/Documents/Repositories/pyxu/doc/guide/lipschitz.ipynb:286: WARNING: invalid syntax (, line 24) in document 'guide/lipschitz' -Parsed source in `python` block: -concatenations: # Access Lipschitz constant of an operator 'op' -lip_const = op.lipschitz - # Estimate Lipschitz constant and update the attribute -op.lipschitz = op.estimate_lipschitz() - # Using trace method -op.lipschitz = op.estimate_lipschitz(method="trace") - # Using SVD method with reduced accuracy -op.lipschitz = op.estimate_lipschitz(method="svd", tol=1e-3) - from pyxu.abc import LinOp -import numpy as np - -rand_op = LinOp.from_array(np.random.random((10000, 10000))) - - rand_op.lipschitz # Unknown as this stage - - get_ipython().run_cell_magic('timeit', '', 'rand_op.lipschitz = rand_op.estimate_lipschitz(method="trace")\n') - - rand_op.lipschitz # Rough estimate - - get_ipython().run_cell_magic('timeit', '', 'rand_op.lipschitz = rand_op.estimate_lipschitz(method="svd", tol=1e-2)\n') - - rand_op.lipschitz # Tighter estimate - -block source: >> op1 = difffunc * linop; - >> op1.lipschitz == difffunc.lipschitz * linop.lipschitz - True - >> op1.diff_lipschitz == difffunc.diff_lipschitz * (linop.lipschitz ** 2) - True - >> op2 = linop * diffmap; - >> op2.diff_lipschitz == linop.lipschitz * diffmap.diff_lipschitz - True - ... +/home/sep/Documents/Repositories/pyxu/doc/guide/lipschitz.ipynb:51: WARNING: File not found: 'api/abc.html#pyxu.abc.LinOp' +/home/sep/Documents/Repositories/pyxu/doc/guide/lipschitz.ipynb:285: WARNING: File not found: 'api/index.html' +/home/sep/Documents/Repositories/pyxu/doc/guide/lipschitz.ipynb:285: WARNING: File not found: 'api/abc/arithmetic.html' /home/sep/Documents/Repositories/pyxu/doc/examples/contributing.ipynb:42: WARNING: invalid syntax (, line 5) in document 'examples/contributing' Parsed source in `python` block: concatenations: try: @@ -99,13 +74,14 @@ concatenations: try: except ImportError as e: print(e) block source: !pip install pyxu-gradient-descent +/home/sep/Documents/Repositories/pyxu/doc/fair/plugins_preview/index.rst:1118: WARNING: toctree contains reference to nonexisting document 'fair/plugins_preview/TokemakRec' /home/sep/Documents/Repositories/pyxu/doc/fair/score.rst:29: WARNING: Bullet list ends without a blank line; unexpected unindent. /home/sep/Documents/Repositories/pyxu/doc/guide/algebra.ipynb:33: WARNING: File not found: 'api/abc.html#pyxu.abc.Map.apply' /home/sep/Documents/Repositories/pyxu/doc/guide/algebra.ipynb:33: WARNING: File not found: 'api/abc.html#pyxu.abc.DiffMap.jacobian' /home/sep/Documents/Repositories/pyxu/doc/guide/algebra.ipynb:33: WARNING: File not found: 'api/abc.html#pyxu.abc.DiffFunc.grad' /home/sep/Documents/Repositories/pyxu/doc/guide/algebra.ipynb:33: WARNING: File not found: 'api/abc.html#pyxu.abc.ProxFunc.prox' /home/sep/Documents/Repositories/pyxu/doc/guide/algebra.ipynb:33: WARNING: File not found: 'api/abc.html#pyxu.abc.LinOp.adjoint' -/home/sep/Documents/Repositories/pyxu/doc/guide/algebra.ipynb:39: WARNING: File not found: 'api/abc/arithmetic.html' +/home/sep/Documents/Repositories/pyxu/doc/guide/algebra.ipynb:39: WARNING: File not found: 'api/abc.html#pyxu-abc-arithmetic' /home/sep/Documents/Repositories/pyxu/doc/guide/algebra.ipynb:41: WARNING: File not found: 'api/abc.html#pyxu.abc.arithmetic.Rule' /home/sep/Documents/Repositories/pyxu/doc/guide/algebra.ipynb:42: WARNING: File not found: 'api/abc.html#pyxu.abc.arithmetic.ScaleRule' /home/sep/Documents/Repositories/pyxu/doc/guide/algebra.ipynb:43: WARNING: File not found: 'api/abc.html#pyxu.abc.arithmetic.ArgScaleRule' @@ -114,68 +90,27 @@ block source: !pip install pyxu-gradient-descent /home/sep/Documents/Repositories/pyxu/doc/guide/algebra.ipynb:46: WARNING: File not found: 'api/abc.html#pyxu.abc.arithmetic.ChainRule' /home/sep/Documents/Repositories/pyxu/doc/guide/algebra.ipynb:47: WARNING: File not found: 'api/abc.html#pyxu.abc.arithmetic.PowerRule' /home/sep/Documents/Repositories/pyxu/doc/guide/algebra.ipynb:48: WARNING: File not found: 'api/abc.html#pyxu.abc.arithmetic.TransposeRule' -/home/sep/Documents/Repositories/pyxu/doc/guide/algebra.ipynb:53: WARNING: File not found: 'api/abc.html#pyxu.abc.DiffFunc' -/home/sep/Documents/Repositories/pyxu/doc/guide/algebra.ipynb:53: WARNING: File not found: 'api/abc.html#pyxu.abc.DiffMap' -/home/sep/Documents/Repositories/pyxu/doc/guide/algebra.ipynb:95: WARNING: File not found: 'api/operator/blocks.html#pyxu.operator.coo_block' -/home/sep/Documents/Repositories/pyxu/doc/guide/algebra.ipynb:95: WARNING: File not found: 'api/operator/blocks.html#pyxu.operator.block' -/home/sep/Documents/Repositories/pyxu/doc/guide/algebra.ipynb:95: WARNING: File not found: 'api/operator/blocks.html#pyxu.operator.block_diag' -/home/sep/Documents/Repositories/pyxu/doc/guide/algebra.ipynb:95: WARNING: File not found: 'api/operator/blocks.html#pyxu.operator.stack' -/home/sep/Documents/Repositories/pyxu/doc/guide/algebra.ipynb:95: WARNING: File not found: 'api/operator/blocks.html#pyxu.operator.vstack' -/home/sep/Documents/Repositories/pyxu/doc/guide/algebra.ipynb:95: WARNING: File not found: 'api/operator/blocks.html#pyxu.operator.hstack' -/home/sep/Documents/Repositories/pyxu/doc/guide/algebra.ipynb:18: WARNING: invalid syntax (, line 1) in document 'guide/algebra' -Parsed source in `python` block: -block source: >> op1 + op2 # Addition of two operators - >> op1 * op2 # Composition of two operators - >> op ** 3 # Exponentiation of an operator - >> op.argscale(c) # Dilation by a scalar 'c' - >> op.argshift(x) # Shifting by a vector 'x' - >> 4 * op # Scaling by a scalar - >> op.T # Transposing -/home/sep/Documents/Repositories/pyxu/doc/guide/algebra.ipynb:55: WARNING: invalid syntax (, line 1) in document 'guide/algebra' -Parsed source in `python` block: -block source: >> h = f * L - >> h.grad(x) = L.jacobian(x).adjoint(f.grad(L(x))) -/home/sep/Documents/Repositories/pyxu/doc/guide/algebra.ipynb:64: WARNING: cannot assign to function call here. Maybe you meant '==' instead of '='? (, line 1) in document 'guide/algebra' -Parsed source in `python` block: -block source: op.apply(arr) = lhs.apply(rhs.apply(arr)) - op.lipschitz = lhs.lipschitz * rhs.lipschitz -/home/sep/Documents/Repositories/pyxu/doc/guide/algebra.ipynb:71: WARNING: cannot assign to function call here. Maybe you meant '==' instead of '='? (, line 1) in document 'guide/algebra' -Parsed source in `python` block: -block source: op.prox(arr, tau) = rhs.adjoint(lhs.prox(rhs.apply(arr), tau)) -/home/sep/Documents/Repositories/pyxu/doc/guide/algebra.ipynb:77: WARNING: cannot assign to function call here. Maybe you meant '==' instead of '='? (, line 1) in document 'guide/algebra' -Parsed source in `python` block: -block source: op.jacobian(arr) = lhs.jacobian(rhs.apply(arr)) * rhs.jacobian(arr) - op.diff_lipschitz = - linear * linear => 0 - linear * diff => lhs.lipschitz * rhs.diff_lipschitz - diff * linear => lhs.diff_lipschitz * (rhs.lipschitz ** 2) - diff * diff => infty -/home/sep/Documents/Repositories/pyxu/doc/guide/algebra.ipynb:88: WARNING: cannot assign to function call here. Maybe you meant '==' instead of '='? (, line 1) in document 'guide/algebra' -Parsed source in `python` block: -block source: op.grad(arr) = lhs.grad(rhs.apply(arr)) @ rhs.jacobian(arr).asarray() -/home/sep/Documents/Repositories/pyxu/doc/guide/algebra.ipynb:100: WARNING: invalid syntax (, line 1) in document 'guide/algebra' -Parsed source in `python` block: -block source: >> coo_block(([A(500,1000), B(1,1000), C(500,500), D(1,3)], # data - ... [[0, 1, 0, 2], # i - ... [0, 0, 2, 1], # j - ]),grid_shape=(3, 3)) +/home/sep/Documents/Repositories/pyxu/doc/guide/algebra.ipynb:50: WARNING: File not found: 'api/abc.html#pyxu.abc.DiffFunc' +/home/sep/Documents/Repositories/pyxu/doc/guide/algebra.ipynb:50: WARNING: File not found: 'api/abc.html#pyxu.abc.DiffMap' +/home/sep/Documents/Repositories/pyxu/doc/guide/algebra.ipynb:59: WARNING: File not found: 'api/operator/blocks.html#pyxu.operator.coo_block' +/home/sep/Documents/Repositories/pyxu/doc/guide/algebra.ipynb:59: WARNING: File not found: 'api/operator/blocks.html#pyxu.operator.block' +/home/sep/Documents/Repositories/pyxu/doc/guide/algebra.ipynb:59: WARNING: File not found: 'api/operator/blocks.html#pyxu.operator.block_diag' +/home/sep/Documents/Repositories/pyxu/doc/guide/algebra.ipynb:59: WARNING: File not found: 'api/operator/blocks.html#pyxu.operator.stack' +/home/sep/Documents/Repositories/pyxu/doc/guide/algebra.ipynb:59: WARNING: File not found: 'api/operator/blocks.html#pyxu.operator.vstack' +/home/sep/Documents/Repositories/pyxu/doc/guide/algebra.ipynb:59: WARNING: File not found: 'api/operator/blocks.html#pyxu.operator.hstack' /home/sep/Documents/Repositories/pyxu/doc/guide/algorithms.ipynb:17: WARNING: File not found: 'api/opt.solver.html#pyxu.opt.solver.CG' /home/sep/Documents/Repositories/pyxu/doc/guide/algorithms.ipynb:29: WARNING: File not found: 'api/opt.solver.html#pyxu.opt.solver.PGD' /home/sep/Documents/Repositories/pyxu/doc/guide/algorithms.ipynb:58: WARNING: File not found: 'api/opt.solver.html#pyxu.opt.solver.CondatVu' /home/sep/Documents/Repositories/pyxu/doc/guide/algorithms.ipynb:58: WARNING: File not found: 'api/opt.solver.html#pyxu.opt.solver.PD3O' /home/sep/Documents/Repositories/pyxu/doc/guide/algorithms.ipynb:58: WARNING: File not found: 'api/opt.solver.html#pyxu.opt.solver.ADMM' /home/sep/Documents/Repositories/pyxu/doc/guide/algorithms.ipynb:58: WARNING: File not found: 'api/opt.solver.html#pyxu.opt.solver.ChambollePock' -/home/sep/Documents/Repositories/pyxu/doc/guide/algorithms.ipynb:74: WARNING: File not found: 'api/opt.solver.html#pyxu.opt.solver.DouglasRachford' -/home/sep/Documents/Repositories/pyxu/doc/guide/algorithms.ipynb:76: WARNING: File not found: 'api/opt.solver.html#pyxu.opt.solver.Adam' -/home/sep/Documents/Repositories/pyxu/doc/guide/algorithms.ipynb:127: WARNING: File not found: 'api/opt.stop.html' -/home/sep/Documents/Repositories/pyxu/doc/guide/algorithms.ipynb:144: WARNING: File not found: 'api/abc.html#pyxu.abc.Solver' -/home/sep/Documents/Repositories/pyxu/doc/guide/algorithms.ipynb:144: WARNING: File not found: 'api/abc.html#pyxu.abc.Solver.m_init' -/home/sep/Documents/Repositories/pyxu/doc/guide/algorithms.ipynb:144: WARNING: File not found: 'api/abc.html#pyxu.abc.Solver.m_step' -/home/sep/Documents/Repositories/pyxu/doc/guide/algorithms.ipynb:148: WARNING: File not found: 'api/abc.html#pyxu.abc.Solver.fit' -/home/sep/Documents/Repositories/pyxu/doc/guide/algorithms.ipynb:66: WARNING: invalid syntax (, line 1) in document 'guide/algorithms' -Parsed source in `python` block: -block source: >> K = vstack([K_1, ..., K_J]) - >> H = hstack([h_1, ..., h_J]) +/home/sep/Documents/Repositories/pyxu/doc/guide/algorithms.ipynb:77: WARNING: File not found: 'api/opt.solver.html#pyxu.opt.solver.DouglasRachford' +/home/sep/Documents/Repositories/pyxu/doc/guide/algorithms.ipynb:79: WARNING: File not found: 'api/opt.solver.html#pyxu.opt.solver.Adam' +/home/sep/Documents/Repositories/pyxu/doc/guide/algorithms.ipynb:131: WARNING: File not found: 'api/opt.stop.html' +/home/sep/Documents/Repositories/pyxu/doc/guide/algorithms.ipynb:148: WARNING: File not found: 'api/abc.html#pyxu.abc.Solver' +/home/sep/Documents/Repositories/pyxu/doc/guide/algorithms.ipynb:148: WARNING: File not found: 'api/abc.html#pyxu.abc.Solver.m_init' +/home/sep/Documents/Repositories/pyxu/doc/guide/algorithms.ipynb:148: WARNING: File not found: 'api/abc.html#pyxu.abc.Solver.m_step' +/home/sep/Documents/Repositories/pyxu/doc/guide/algorithms.ipynb:152: WARNING: File not found: 'api/abc.html#pyxu.abc.Solver.fit' /home/sep/Documents/Repositories/pyxu/doc/guide/diff_ops.ipynb:11: WARNING: File not found: 'api/operator/linop.html#derivatives' /home/sep/Documents/Repositories/pyxu/doc/guide/diff_ops.ipynb:18: WARNING: File not found: 'api/operator/linop.html#pyxu.operator.PartialDerivative' /home/sep/Documents/Repositories/pyxu/doc/guide/diff_ops.ipynb:535: WARNING: File not found: 'api/operator/linop.html#pyxu.operator.Gradient' @@ -546,12 +481,25 @@ block source: def Hessian( parallel: bool = False, **diff_kwargs, ) -> pyct.OpT: +/home/sep/Documents/Repositories/pyxu/doc/intro/tomo.ipynb:284: WARNING: File not found: 'api/abc.html#pyxu.abc.Operator.expr' +/home/sep/Documents/Repositories/pyxu/doc/intro/tomo.ipynb:451: WARNING: File not found: 'api/abc.html#pyxu.abc.LinOp.pinv' +/home/sep/Documents/Repositories/pyxu/doc/intro/tomo.ipynb:451: WARNING: File not found: 'api/abc.html#pyxu.abc.LinOp' +/home/sep/Documents/Repositories/pyxu/doc/intro/tomo.ipynb:666: WARNING: File not found: 'api/operator/linop.html#pyxu.operator.Gradient' +/home/sep/Documents/Repositories/pyxu/doc/intro/tomo.ipynb:668: WARNING: File not found: 'api/abc.html#pyxu.abc.ProxFunc.moreau_envelope' +/home/sep/Documents/Repositories/pyxu/doc/intro/tomo.ipynb:677: WARNING: File not found: 'api/operator/func.html#pyxu.operator.PositiveL1Norm' +/home/sep/Documents/Repositories/pyxu/doc/intro/tomo.ipynb:686: WARNING: File not found: 'api/operator/func.html#pyxu.operator.SquaredL2Norm' +/home/sep/Documents/Repositories/pyxu/doc/intro/tomo.ipynb:687: WARNING: File not found: 'api/abc.html#pyxu.abc.Func.asloss' +/home/sep/Documents/Repositories/pyxu/doc/intro/tomo.ipynb:699: WARNING: File not found: 'api/abc.html#pyxu.abc.DiffMap.estimate_diff_lipschitz' +/home/sep/Documents/Repositories/pyxu/doc/intro/tomo.ipynb:914: WARNING: File not found: 'api/abc.html#pyxu.abc.ProxDiffFunc' +/home/sep/Documents/Repositories/pyxu/doc/fair/plugins_preview/TokamakRec.rst: WARNING: document isn't included in any toctree /home/sep/Documents/Repositories/pyxu/doc/references.rst:5: WARNING: Citation [ProxAlg] is not referenced. /home/sep/Documents/Repositories/pyxu/doc/references.rst:8: WARNING: Citation [ProxSplit] is not referenced. /home/sep/Documents/Repositories/pyxu/doc/references.rst:19: WARNING: Citation [P2] is not referenced. /home/sep/Documents/Repositories/pyxu/doc/references.rst:20: WARNING: Citation [GaussProcesses] is not referenced. /home/sep/Documents/Repositories/pyxu/doc/references.rst:21: WARNING: Citation [SubGauss] is not referenced. /home/sep/Documents/Repositories/pyxu/doc/references.rst:29: WARNING: Citation [UQ_MCMC] is not referenced. +/home/sep/Documents/Repositories/pyxu/src/pyxu/operator/interop/jax.py:docstring of pyxu.operator.interop.jax._from_jax:: WARNING: py:class reference target not found: pyxu.operator.interop.jax.JaxArray +/home/sep/Documents/Repositories/pyxu/src/pyxu/operator/interop/jax.py:docstring of pyxu.operator.interop.jax._to_jax:: WARNING: py:class reference target not found: pyxu.operator.interop.jax.JaxArray /home/sep/Documents/Repositories/pyxu/src/pyxu/operator/blocks.py:docstring of pyxu.operator.blocks.coo_block:33: WARNING: py:mod reference target not found: pyxu.operator.blocks /home/sep/Documents/Repositories/pyxu/src/pyxu/operator/linop/pad.py:docstring of pyxu.operator.linop.pad.Pad.__init__:10: WARNING: py:class reference target not found: pyxu.operator.linop.pad.Pad.WidthSpec /home/sep/Documents/Repositories/pyxu/src/pyxu/operator/linop/filter.py:docstring of pyxu.operator.linop.filter.MovingAverage:30: WARNING: py:class reference target not found: pyxu.operator.linop.filter.IndexSpec diff --git a/_downloads/05ca4c1d7c08843dd63c373f56165367/linop-11_00.hires.png b/_downloads/05ca4c1d7c08843dd63c373f56165367/linop-11_00.hires.png index 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L1Norm(ProxFunc): - def __init__(self, dim: int): - super().__init__(shape=(1, dim)) - def apply(self, arr): - return np.linalg.norm(arr, axis=-1, keepdims=True, ord=1) - def prox(self, arr, tau): - return np.clip(np.abs(arr)-tau, a_min=0, a_max=None) * np.sign(arr) - -N = 512 -l1_norm = L1Norm(dim=N) -mus = [0.1, 0.5, 1] -smooth_l1_norms = [l1_norm.moreau_envelope(mu) for mu in mus] - -x = np.linspace(-1, 1, N)[:, None] -labels=['mu=0'] -labels.extend([f'mu={mu}' for mu in mus]) -plt.figure() -plt.plot(x, l1_norm(x)) -for f in smooth_l1_norms: - plt.plot(x, f(x)) -plt.legend(labels) -plt.title('Moreau Envelope') - -labels=[f'mu={mu}' for mu in mus] -plt.figure() -for f in smooth_l1_norms: - plt.plot(x, f.grad(x)) -plt.legend(labels) -plt.title('Derivative of Moreau Envelope') \ No newline at end of file diff --git a/_downloads/382f1aebcc11e28238e392f33afc5648/misc-2.py b/_downloads/382f1aebcc11e28238e392f33afc5648/misc-2.py deleted file mode 100644 index 2ebee297..00000000 --- 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    Returns:
    @@ -2201,12 +2201,12 @@

    pyxu.abc.solverReturns:

    • data (Mapping) – Value(s) of log_var (s) after last iteration.

    • -
    • history (numpy.ndarray, None) – (N_iter,) records of stopping-criteria values per iteration.

    • +
    • history (numpy.ndarray, None) – (N_iter,) records of stopping-criteria values per iteration.

    Return type:
    -

    tuple[dict[str, Real | NDArray | None], ndarray | None]

    +

    tuple[dict[str, Real | NDArray | None], ndarray | None]

    Notes

    @@ -2601,13 +2601,13 @@

    pyxu.abc.solver diff --git a/api/abc/arithmetic.html b/api/abc/arithmetic.html deleted file mode 100644 index f6d68d97..00000000 --- a/api/abc/arithmetic.html +++ /dev/null @@ -1,1060 +0,0 @@ - - - - - - - - - - - pyxu.abc.arithmetic — Pyxu Documentation - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
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    -

    pyxu.abc.arithmetic#

    -
    -
    -class Rule[source]#
    -

    Bases: object

    -

    General arithmetic rule.

    -

    This class defines default arithmetic rules applicable unless re-defined by sub-classes.

    -
    -
    -op()[source]#
    -
    -
    Returns:
    -

    op – Synthesize operator.

    -
    -
    Return type:
    -

    OpT

    -
    -
    -
    - -
    - -
    -
    -class ScaleRule(op, cst)[source]#
    -

    Bases: Rule

    -

    Arithmetic rules for element-wise scaling: \(B(x) = \alpha A(x)\).

    -

    Special Cases:

    -
    \alpha = 0  => NullOp/NullFunc
    -\alpha = 1  => self
    -
    -
    -

    Else:

    -
    |--------------------------|-------------|--------------------------------------------------------------------|
    -|         Property         |  Preserved? |                     Arithmetic Update Rule(s)                      |
    -|--------------------------|-------------|--------------------------------------------------------------------|
    -| CAN_EVAL                 | yes         | op_new.apply(arr) = op_old.apply(arr) * \alpha                     |
    -|                          |             | op_new.lipschitz = op_old.lipschitz * abs(\alpha)                  |
    -|--------------------------|-------------|--------------------------------------------------------------------|
    -| FUNCTIONAL               | yes         | op_new.asloss(\beta) = op_old.asloss(\beta) * \alpha               |
    -|--------------------------|-------------|--------------------------------------------------------------------|
    -| PROXIMABLE               | \alpha > 0  | op_new.prox(arr, tau) = op_old.prox(arr, tau * \alpha)             |
    -|--------------------------|-------------|--------------------------------------------------------------------|
    -| DIFFERENTIABLE           | yes         | op_new.jacobian(arr) = op_old.jacobian(arr) * \alpha               |
    -|                          |             | op_new.diff_lipschitz = op_old.diff_lipschitz * abs(\alpha)        |
    -|--------------------------|-------------|--------------------------------------------------------------------|
    -| DIFFERENTIABLE_FUNCTION  | yes         | op_new.grad(arr) = op_old.grad(arr) * \alpha                       |
    -|--------------------------|-------------|--------------------------------------------------------------------|
    -| QUADRATIC                | \alpha > 0  | Q, c, t = op_old._quad_spec()                                      |
    -|                          |             | op_new._quad_spec() = (\alpha * Q, \alpha * c, \alpha * t)         |
    -|--------------------------|-------------|--------------------------------------------------------------------|
    -| LINEAR                   | yes         | op_new.adjoint(arr) = op_old.adjoint(arr) * \alpha                 |
    -|                          |             | op_new.asarray() = op_old.asarray() * \alpha                       |
    -|                          |             | op_new.svdvals() = op_old.svdvals() * abs(\alpha)                  |
    -|                          |             | op_new.pinv(x, damp) = op_old.pinv(x, damp / (\alpha**2)) / \alpha |
    -|                          |             | op_new.gram() = op_old.gram() * (\alpha**2)                        |
    -|                          |             | op_new.cogram() = op_old.cogram() * (\alpha**2)                    |
    -|--------------------------|-------------|--------------------------------------------------------------------|
    -| LINEAR_SQUARE            | yes         | op_new.trace() = op_old.trace() * \alpha                           |
    -|--------------------------|-------------|--------------------------------------------------------------------|
    -| LINEAR_NORMAL            | yes         |                                                                    |
    -|--------------------------|-------------|--------------------------------------------------------------------|
    -| LINEAR_UNITARY           | \alpha = -1 |                                                                    |
    -|--------------------------|-------------|--------------------------------------------------------------------|
    -| LINEAR_SELF_ADJOINT      | yes         |                                                                    |
    -|--------------------------|-------------|--------------------------------------------------------------------|
    -| LINEAR_POSITIVE_DEFINITE | \alpha > 0  |                                                                    |
    -|--------------------------|-------------|--------------------------------------------------------------------|
    -| LINEAR_IDEMPOTENT        | no          |                                                                    |
    -|--------------------------|-------------|--------------------------------------------------------------------|
    -
    -
    -
    -
    Parameters:
    -
    -
    -
    -
    - -
    -
    -class ArgScaleRule(op, cst)[source]#
    -

    Bases: Rule

    -

    Arithmetic rules for element-wise parameter scaling: \(B(x) = A(\alpha x)\).

    -

    Special Cases:

    -
    \alpha = 0  => ConstantValued (w/ potential vector-valued output)
    -\alpha = 1  => self
    -
    -
    -

    Else:

    -
    |--------------------------|-------------|-----------------------------------------------------------------------------|
    -|         Property         |  Preserved? |                          Arithmetic Update Rule(s)                          |
    -|--------------------------|-------------|-----------------------------------------------------------------------------|
    -| CAN_EVAL                 | yes         | op_new.apply(arr) = op_old.apply(arr * \alpha)                              |
    -|                          |             | op_new.lipschitz = op_old.lipschitz * abs(\alpha)                           |
    -|--------------------------|-------------|-----------------------------------------------------------------------------|
    -| FUNCTIONAL               | yes         | op_new.asloss(\beta) = AMBIGUOUS -> DISABLED                                |
    -|--------------------------|-------------|-----------------------------------------------------------------------------|
    -| PROXIMABLE               | yes         | op_new.prox(arr, tau) = op_old.prox(\alpha * arr, \alpha**2 * tau) / \alpha |
    -|--------------------------|-------------|-----------------------------------------------------------------------------|
    -| DIFFERENTIABLE           | yes         | op_new.diff_lipschitz = op_old.diff_lipschitz * (\alpha**2)                 |
    -|                          |             | op_new.jacobian(arr) = op_old.jacobian(arr * \alpha) * \alpha               |
    -|--------------------------|-------------|-----------------------------------------------------------------------------|
    -| DIFFERENTIABLE_FUNCTION  | yes         | op_new.grad(arr) = op_old.grad(\alpha * arr) * \alpha                       |
    -|--------------------------|-------------|-----------------------------------------------------------------------------|
    -| QUADRATIC                | yes         | Q, c, t = op_old._quad_spec()                                               |
    -|                          |             | op_new._quad_spec() = (\alpha**2 * Q, \alpha * c, t)                        |
    -|--------------------------|-------------|-----------------------------------------------------------------------------|
    -| LINEAR                   | yes         | op_new.adjoint(arr) = op_old.adjoint(arr) * \alpha                          |
    -|                          |             | op_new.asarray() = op_old.asarray() * \alpha                                |
    -|                          |             | op_new.svdvals() = op_old.svdvals() * abs(\alpha)                           |
    -|                          |             | op_new.pinv(x, damp) = op_old.pinv(x, damp / (\alpha**2)) / \alpha          |
    -|                          |             | op_new.gram() = op_old.gram() * (\alpha**2)                                 |
    -|                          |             | op_new.cogram() = op_old.cogram() * (\alpha**2)                             |
    -|--------------------------|-------------|-----------------------------------------------------------------------------|
    -| LINEAR_SQUARE            | yes         | op_new.trace() = op_old.trace() * \alpha                                    |
    -|--------------------------|-------------|-----------------------------------------------------------------------------|
    -| LINEAR_NORMAL            | yes         |                                                                             |
    -|--------------------------|-------------|-----------------------------------------------------------------------------|
    -| LINEAR_UNITARY           | \alpha = -1 |                                                                             |
    -|--------------------------|-------------|-----------------------------------------------------------------------------|
    -| LINEAR_SELF_ADJOINT      | yes         |                                                                             |
    -|--------------------------|-------------|-----------------------------------------------------------------------------|
    -| LINEAR_POSITIVE_DEFINITE | \alpha > 0  |                                                                             |
    -|--------------------------|-------------|-----------------------------------------------------------------------------|
    -| LINEAR_IDEMPOTENT        | no          |                                                                             |
    -|--------------------------|-------------|-----------------------------------------------------------------------------|
    -
    -
    -
    -
    Parameters:
    -
    -
    -
    -
    - -
    -
    -class ArgShiftRule(op, cst)[source]#
    -

    Bases: Rule

    -

    Arithmetic rules for parameter shifting: \(B(x) = A(x - c)\).

    -

    Special Cases:

    -
    \shift = 0  => self
    -
    -
    -

    Else:

    -
    |--------------------------|------------|-----------------------------------------------------------------|
    -|         Property         | Preserved? |                    Arithmetic Update Rule(s)                    |
    -|--------------------------|------------|-----------------------------------------------------------------|
    -| CAN_EVAL                 | yes        | op_new.apply(arr) = op_old.apply(arr + \shift)                  |
    -|                          |            | op_new.lipschitz = op_old.lipschitz                             |
    -|--------------------------|------------|-----------------------------------------------------------------|
    -| FUNCTIONAL               | yes        | op_new.asloss(\beta) = AMBIGUOUS -> DISABLED                    |
    -|--------------------------|------------|-----------------------------------------------------------------|
    -| PROXIMABLE               | yes        | op_new.prox(arr, tau) = op_old.prox(arr + \shift, tau) - \shift |
    -|--------------------------|------------|-----------------------------------------------------------------|
    -| DIFFERENTIABLE           | yes        | op_new.diff_lipschitz = op_old.diff_lipschitz                   |
    -|                          |            | op_new.jacobian(arr) = op_old.jacobian(arr + \shift)            |
    -|--------------------------|------------|-----------------------------------------------------------------|
    -| DIFFERENTIABLE_FUNCTION  | yes        | op_new.grad(arr) = op_old.grad(arr + \shift)                    |
    -|--------------------------|------------|-----------------------------------------------------------------|
    -| QUADRATIC                | yes        | Q, c, t = op_old._quad_spec()                                   |
    -|                          |            | op_new._quad_spec() = (Q, c + Q @ \shift, op_old.apply(\shift)) |
    -|--------------------------|------------|-----------------------------------------------------------------|
    -| LINEAR                   | no         |                                                                 |
    -|--------------------------|------------|-----------------------------------------------------------------|
    -| LINEAR_SQUARE            | no         |                                                                 |
    -|--------------------------|------------|-----------------------------------------------------------------|
    -| LINEAR_NORMAL            | no         |                                                                 |
    -|--------------------------|------------|-----------------------------------------------------------------|
    -| LINEAR_UNITARY           | no         |                                                                 |
    -|--------------------------|------------|-----------------------------------------------------------------|
    -| LINEAR_SELF_ADJOINT      | no         |                                                                 |
    -|--------------------------|------------|-----------------------------------------------------------------|
    -| LINEAR_POSITIVE_DEFINITE | no         |                                                                 |
    -|--------------------------|------------|-----------------------------------------------------------------|
    -| LINEAR_IDEMPOTENT        | no         |                                                                 |
    -|--------------------------|------------|-----------------------------------------------------------------|
    -
    -
    -
    -
    Parameters:
    -
    -
    -
    -
    - -
    -
    -class AddRule(lhs, rhs)[source]#
    -

    Bases: Rule

    -

    Arithmetic rules for operator addition: \(C(x) = A(x) + B(x)\).

    -

    The output type of AddRule(A.squeeze(), B.squeeze()) is summarized in the table below (LHS/RHS -commute):

    -
    |---------------|-----|------|---------|----------|----------|--------------|-----------|---------|--------------|------------|------------|------------|---------------|---------------|------------|---------------|
    -|   LHS / RHS   | Map | Func | DiffMap | DiffFunc | ProxFunc | ProxDiffFunc | Quadratic |  LinOp  |   LinFunc    |  SquareOp  |  NormalOp  |   UnitOp   | SelfAdjointOp |    PosDefOp   |   ProjOp   |   OrthProjOp  |
    -|---------------|-----|------|---------|----------|----------|--------------|-----------|---------|--------------|------------|------------|------------|---------------|---------------|------------|---------------|
    -| Map           | Map | Map  | Map     | Map      | Map      | Map          | Map       | Map     | Map          | Map        | Map        | Map        | Map           | Map           | Map        | Map           |
    -| Func          |     | Func | Map     | Func     | Func     | Func         | Func      | Map     | Func         | Map        | Map        | Map        | Map           | Map           | Map        | Map           |
    -| DiffMap       |     |      | DiffMap | DiffMap  | Map      | DiffMap      | DiffMap   | DiffMap | DiffMap      | DiffMap    | DiffMap    | DiffMap    | DiffMap       | DiffMap       | DiffMap    | DiffMap       |
    -| DiffFunc      |     |      |         | DiffFunc | Func     | DiffFunc     | DiffFunc  | DiffMap | DiffFunc     | DiffMap    | DiffMap    | DiffMap    | DiffMap       | DiffMap       | DiffMap    | DiffMap       |
    -| ProxFunc      |     |      |         |          | Func     | Func         | Func      | Map     | ProxFunc     | Map        | Map        | Map        | Map           | Map           | Map        | Map           |
    -| ProxDiffFunc  |     |      |         |          |          | DiffFunc     | DiffFunc  | DiffMap | ProxDiffFunc | DiffMap    | DiffMap    | DiffMap    | DiffMap       | DiffMap       | DiffMap    | DiffMap       |
    -| Quadratic     |     |      |         |          |          |              | Quadratic | DiffMap | Quadratic    | DiffMap    | DiffMap    | DiffMap    | DiffMap       | DiffMap       | DiffMap    | DiffMap       |
    -| LinOp         |     |      |         |          |          |              |           | LinOp   | LinOp        | IMPOSSIBLE | IMPOSSIBLE | IMPOSSIBLE | IMPOSSIBLE    | IMPOSSIBLE    | IMPOSSIBLE | IMPOSSIBLE    |
    -| LinFunc       |     |      |         |          |          |              |           |         | LinFunc      | SquareOp   | SquareOp   | SquareOp   | SquareOp      | SquareOp      | SquareOp   | SquareOp      |
    -| SquareOp      |     |      |         |          |          |              |           |         |              | SquareOp   | SquareOp   | SquareOp   | SquareOp      | SquareOp      | SquareOp   | SquareOp      |
    -| NormalOp      |     |      |         |          |          |              |           |         |              |            | SquareOp   | SquareOp   | SquareOp      | SquareOp      | SquareOp   | SquareOp      |
    -| UnitOp        |     |      |         |          |          |              |           |         |              |            |            | SquareOp   | SquareOp      | SquareOp      | SquareOp   | SquareOp      |
    -| SelfAdjointOp |     |      |         |          |          |              |           |         |              |            |            |            | SelfAdjointOp | SelfAdjointOp | SquareOp   | SelfAdjointOp |
    -| PosDefOp      |     |      |         |          |          |              |           |         |              |            |            |            |               | PosDefOp      | SquareOp   | PosDefOp      |
    -| ProjOp        |     |      |         |          |          |              |           |         |              |            |            |            |               |               | SquareOp   | SquareOp      |
    -| OrthProjOp    |     |      |         |          |          |              |           |         |              |            |            |            |               |               |            | SelfAdjointOp |
    -|---------------|-----|------|---------|----------|----------|--------------|-----------|---------|--------------|------------|------------|------------|---------------|---------------|------------|---------------|
    -
    -
    -

    Arithmetic Update Rule(s):

    -
    * CAN_EVAL
    -    op.apply(arr) = _lhs.apply(arr) + _rhs.apply(arr)
    -    op.lipschitz = _lhs.lipschitz + _rhs.lipschitz
    -    IMPORTANT: if range-broadcasting takes place (ex: LHS(1,) + RHS(M,)), then the broadcasted
    -               operand's Lipschitz constant must be magnified by \sqrt{M}.
    -
    -* FUNCTIONAL
    -    op.asloss(\beta) = _lhs.asloss(\beta) + _rhs.asloss(\beta)
    -                       may be ambiguous -> warning
    -
    -* PROXIMABLE
    -    op.prox(arr, tau) = _lhs.prox(arr - tau * _rhs.grad(arr), tau)
    -                  OR  = _rhs.prox(arr - tau * _lhs.grad(arr), tau)
    -        IMPORTANT: the one calling .grad() should be either (lhs, rhs) which has LINEAR property
    -
    -* DIFFERENTIABLE
    -    op.jacobian(arr) = _lhs.jacobian(arr) + _rhs.jacobian(arr)
    -    op.diff_lipschitz = _lhs.diff_lipschitz + _rhs.diff_lipschitz
    -    IMPORTANT: if range-broadcasting takes place (ex: LHS(1,) + RHS(M,)), then the broadcasted
    -               operand's diff-Lipschitz constant must be magnified by \sqrt{M}.
    -
    -* DIFFERENTIABLE_FUNCTION
    -    op.grad(arr) = _lhs.grad(arr) + _rhs.grad(arr)
    -
    -* LINEAR
    -    op.adjoint(arr) = _lhs.adjoint(arr) + _rhs.adjoint(arr)
    -    IMPORTANT: if range-broadcasting takes place (ex: LHS(1,) + RHS(M,)), then the broadcasted
    -               operand's adjoint-input must be averaged.
    -    op.asarray() = _lhs.asarray() + _rhs.asarray()
    -    op.gram() = _lhs.gram() + _rhs.gram() + (_lhs.T * _rhs) + (_rhs.T * _lhs)
    -    op.cogram() = _lhs.cogram() + _rhs.cogram() + (_lhs * _rhs.T) + (_rhs * _lhs.T)
    -
    -* LINEAR_SQUARE
    -    op.trace() = _lhs.trace() + _rhs.trace()
    -
    -* QUADRATIC
    -    lhs = rhs = quadratic
    -      Q_l, c_l, t_l = lhs._quad_spec()
    -      Q_r, c_r, t_r = rhs._quad_spec()
    -      op._quad_spec() = (Q_l + Q_r, c_l + c_r, t_l + t_r)
    -    lhs, rhs = quadratic, linear
    -      Q, c, t = lhs._quad_spec()
    -      op._quad_spec() = (Q, c + rhs, t)
    -
    -
    -
    -
    Parameters:
    -
      -
    • lhs (OpT) –

    • -
    • rhs (OpT) –

    • -
    -
    -
    -
    - -
    -
    -class ChainRule(lhs, rhs)[source]#
    -

    Bases: Rule

    -

    Arithmetic rules for operator composition: \(C(x) = (A \circ B)(x)\).

    -

    The output type of ChainRule(A.squeeze(), B.squeeze()) is summarized in the table below:

    -
    |---------------|------|------------|----------|------------|------------|----------------|----------------------|------------------|------------|-----------|-----------|--------------|---------------|-----------|-----------|------------|
    -|   LHS / RHS   | Map  |    Func    | DiffMap  |  DiffFunc  |  ProxFunc  |  ProxDiffFunc  |      Quadratic       |      LinOp       |  LinFunc   |  SquareOp |  NormalOp |    UnitOp    | SelfAdjointOp |  PosDefOp |   ProjOp  | OrthProjOp |
    -|---------------|------|------------|----------|------------|------------|----------------|----------------------|------------------|------------|-----------|-----------|--------------|---------------|-----------|-----------|------------|
    -| Map           | Map  | Map        | Map      | Map        | Map        | Map            | Map                  | Map              | Map        | Map       | Map       | Map          | Map           | Map       | Map       | Map        |
    -| Func          | Func | Func       | Func     | Func       | Func       | Func           | Func                 | Func             | Func       | Func      | Func      | Func         | Func          | Func      | Func      | Func       |
    -| DiffMap       | Map  | Map        | DiffMap  | DiffMap    | Map        | DiffMap        | DiffMap              | DiffMap          | DiffMap    | DiffMap   | DiffMap   | DiffMap      | DiffMap       | DiffMap   | DiffMap   | DiffMap    |
    -| DiffFunc      | Func | Func       | DiffFunc | DiffFunc   | Func       | DiffFunc       | DiffFunc             | DiffFunc         | DiffFunc   | DiffFunc  | DiffFunc  | DiffFunc     | DiffFunc      | DiffFunc  | DiffFunc  | DiffFunc   |
    -| ProxFunc      | Func | Func       | Func     | Func       | Func       | Func           | Func                 | Func             | Func       | Func      | Func      | ProxFunc     | Func          | Func      | Func      | Func       |
    -| ProxDiffFunc  | Func | Func       | DiffFunc | DiffFunc   | Func       | DiffFunc       | DiffFunc             | DiffFunc         | DiffFunc   | DiffFunc  | DiffFunc  | ProxDiffFunc | DiffFunc      | DiffFunc  | DiffFunc  | DiffFunc   |
    -| Quadratic     | Func | Func       | DiffFunc | DiffFunc   | Func       | DiffFunc       | DiffFunc             | Quadratic        | Quadratic  | Quadratic | Quadratic | Quadratic    | Quadratic     | Quadratic | Quadratic | Quadratic  |
    -| LinOp         | Map  | Func       | DiffMap  | DiffMap    | Map        | DiffMap        | DiffMap              | LinOp / SquareOp | LinOp      | LinOp     | LinOp     | LinOp        | LinOp         | LinOp     | LinOp     | LinOp      |
    -| LinFunc       | Func | Func       | DiffFunc | DiffFunc   | [Prox]Func | [Prox]DiffFunc | DiffFunc / Quadratic | LinFunc          | LinFunc    | LinFunc   | LinFunc   | LinFunc      | LinFunc       | LinFunc   | LinFunc   | LinFunc    |
    -| SquareOp      | Map  | IMPOSSIBLE | DiffMap  | IMPOSSIBLE | IMPOSSIBLE | IMPOSSIBLE     | IMPOSSIBLE           | LinOp            | IMPOSSIBLE | SquareOp  | SquareOp  | SquareOp     | SquareOp      | SquareOp  | SquareOp  | SquareOp   |
    -| NormalOp      | Map  | IMPOSSIBLE | DiffMap  | IMPOSSIBLE | IMPOSSIBLE | IMPOSSIBLE     | IMPOSSIBLE           | LinOp            | IMPOSSIBLE | SquareOp  | SquareOp  | SquareOp     | SquareOp      | SquareOp  | SquareOp  | SquareOp   |
    -| UnitOp        | Map  | IMPOSSIBLE | DiffMap  | IMPOSSIBLE | IMPOSSIBLE | IMPOSSIBLE     | IMPOSSIBLE           | LinOp            | IMPOSSIBLE | SquareOp  | SquareOp  | UnitOp       | SquareOp      | SquareOp  | SquareOp  | SquareOp   |
    -| SelfAdjointOp | Map  | IMPOSSIBLE | DiffMap  | IMPOSSIBLE | IMPOSSIBLE | IMPOSSIBLE     | IMPOSSIBLE           | LinOp            | IMPOSSIBLE | SquareOp  | SquareOp  | SquareOp     | SquareOp      | SquareOp  | SquareOp  | SquareOp   |
    -| PosDefOp      | Map  | IMPOSSIBLE | DiffMap  | IMPOSSIBLE | IMPOSSIBLE | IMPOSSIBLE     | IMPOSSIBLE           | LinOp            | IMPOSSIBLE | SquareOp  | SquareOp  | SquareOp     | SquareOp      | SquareOp  | SquareOp  | SquareOp   |
    -| ProjOp        | Map  | IMPOSSIBLE | DiffMap  | IMPOSSIBLE | IMPOSSIBLE | IMPOSSIBLE     | IMPOSSIBLE           | LinOp            | IMPOSSIBLE | SquareOp  | SquareOp  | SquareOp     | SquareOp      | SquareOp  | SquareOp  | SquareOp   |
    -| OrthProjOp    | Map  | IMPOSSIBLE | DiffMap  | IMPOSSIBLE | IMPOSSIBLE | IMPOSSIBLE     | IMPOSSIBLE           | LinOp            | IMPOSSIBLE | SquareOp  | SquareOp  | SquareOp     | SquareOp      | SquareOp  | SquareOp  | SquareOp   |
    -|---------------|------|------------|----------|------------|------------|----------------|----------------------|------------------|------------|-----------|-----------|--------------|---------------|-----------|-----------|------------|
    -
    -
    -

    Arithmetic Update Rule(s):

    -
    * CAN_EVAL
    -    op.apply(arr) = _lhs.apply(_rhs.apply(arr))
    -    op.lipschitz = _lhs.lipschitz * _rhs.lipschitz
    -
    -* FUNCTIONAL
    -    op.asloss(\beta) = ambiguous -> disabled
    -
    -* PROXIMABLE (RHS Unitary only)
    -    op.prox(arr, tau) = _rhs.adjoint(_lhs.prox(_rhs.apply(arr), tau))
    -
    -* DIFFERENTIABLE
    -    op.jacobian(arr) = _lhs.jacobian(_rhs.apply(arr)) * _rhs.jacobian(arr)
    -    op.diff_lipschitz =
    -        quadratic            => _quad_spec().Q.lipschitz
    -        linear \comp linear  => 0
    -        linear \comp diff    => _lhs.lipschitz * _rhs.diff_lipschitz
    -        diff   \comp linear  => _lhs.diff_lipschitz * (_rhs.lipschitz ** 2)
    -        diff   \comp diff    => \infty
    -
    -* DIFFERENTIABLE_FUNCTION (1D input)
    -    op.grad(arr) = _lhs.grad(_rhs.apply(arr)) @ _rhs.jacobian(arr).asarray()
    -
    -* LINEAR
    -    op.adjoint(arr) = _rhs.adjoint(_lhs.adjoint(arr))
    -    op.asarray() = _lhs.asarray() @ _rhs.asarray()
    -    op.gram() = _rhs.T @ _lhs.gram() @ _rhs
    -    op.cogram() = _lhs @ _rhs.cogram() @ _lhs.T
    -
    -* QUADRATIC
    -    Q, c, t = _lhs._quad_spec()
    -    op._quad_spec() = (_rhs.T * Q * _rhs, _rhs.T * c, t)
    -
    -
    -
    -
    Parameters:
    -
      -
    • lhs (OpT) –

    • -
    • rhs (OpT) –

    • -
    -
    -
    -
    - -
    -
    -class PowerRule(op, k)[source]#
    -

    Bases: Rule

    -

    Arithmetic rules for operator exponentiation: \(B(x) = A^{k}(x)\).

    -

    Special Cases:

    -
    k = 0  => IdentityOp
    -
    -
    -

    Else:

    -
    B = A \circ ... \circ A  (k-1 compositions)
    -
    -
    -
    -
    Parameters:
    -
    -
    -
    -
    - -
    -
    -class TransposeRule(op)[source]#
    -

    Bases: Rule

    -

    Arithmetic rules for LinOp transposition: \(B(x) = A^{T}(x)\).

    -

    Arithmetic Update Rule(s):

    -
    * CAN_EVAL
    -    opT.apply(arr) = op.adjoint(arr)
    -    opT.lipschitz = op.lipschitz
    -
    -* FUNCTIONAL
    -    opT.asloss(\beta) = UNDEFINED
    -
    -* PROXIMABLE
    -    opT.prox(arr, tau) = LinFunc.prox(arr, tau)
    -
    -* DIFFERENTIABLE
    -    opT.jacobian(arr) = opT
    -    opT.diff_lipschitz = 0
    -
    -* DIFFERENTIABLE_FUNCTION
    -    opT.grad(arr) = LinFunc.grad(arr)
    -
    -* LINEAR
    -    opT.adjoint(arr) = op.apply(arr)
    -    opT.asarray() = op.asarray().T
    -    opT.gram() = op.cogram()
    -    opT.cogram() = op.gram()
    -    opT.svdvals() = op.svdvals()
    -
    -* LINEAR_SQUARE
    -    opT.trace() = op.trace()
    -
    -
    -
    -
    Parameters:
    -

    op (OpT) –

    -
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    - - \ No newline at end of file diff --git a/api/abc/index.html b/api/abc/index.html deleted file mode 100644 index c22640a8..00000000 --- a/api/abc/index.html +++ /dev/null @@ -1,621 +0,0 @@ - - - - - - - - - - - pyxu.abc — Pyxu Documentation - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
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    - -
    -

    pyxu.abc#

    -
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    - - \ No newline at end of file diff --git a/api/abc/operator.html b/api/abc/operator.html deleted file mode 100644 index 52ca7c9c..00000000 --- a/api/abc/operator.html +++ /dev/null @@ -1,2130 +0,0 @@ - - - - - - - - - - - pyxu.abc.operator — Pyxu Documentation - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
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    -

    pyxu.abc.operator#

    -
    -
    -class Property(value, names=None, *, module=None, qualname=None, type=None, start=1, boundary=None)[source]#
    -

    Bases: Enum

    -

    Mathematical property.

    -
    -

    See also

    -

    Operator

    -
    -
    -
    -CAN_EVAL = 1#
    -
    - -
    -
    -FUNCTIONAL = 2#
    -
    - -
    -
    -PROXIMABLE = 3#
    -
    - -
    -
    -DIFFERENTIABLE = 4#
    -
    - -
    -
    -DIFFERENTIABLE_FUNCTION = 5#
    -
    - -
    -
    -LINEAR = 6#
    -
    - -
    -
    -LINEAR_SQUARE = 7#
    -
    - -
    -
    -LINEAR_NORMAL = 8#
    -
    - -
    -
    -LINEAR_IDEMPOTENT = 9#
    -
    - -
    -
    -LINEAR_SELF_ADJOINT = 10#
    -
    - -
    -
    -LINEAR_POSITIVE_DEFINITE = 11#
    -
    - -
    -
    -LINEAR_UNITARY = 12#
    -
    - -
    -
    -QUADRATIC = 13#
    -
    - -
    -
    -arithmetic_methods()[source]#
    -

    Instance methods affected by arithmetic operations.

    -
    -
    Return type:
    -

    Set[str]

    -
    -
    -
    - -
    - -
    -
    -class Operator(shape)[source]#
    -

    Bases: object

    -

    Abstract Base Class for Pyxu operators.

    -

    Goals:

    -
      -
    • enable operator arithmetic.

    • -
    • cast operators to specialized forms.

    • -
    • attach Property tags encoding certain mathematical properties. Each core sub-class must -have a unique set of properties to be distinguishable from its peers.

    • -
    -
    -
    Parameters:
    -

    shape (tuple[Integral, Integral]) –

    -
    -
    -
    -
    -__init__(shape)[source]#
    -
    -
    Parameters:
    -

    shape (OpShape) – (N, M) operator shape.

    -
    -
    -
    - -
    -
    -property shape: tuple[Integral, Integral]#
    -

    Return (N, M) operator shape.

    -
    - -
    -
    -property dim: Integral#
    -

    Return dimension of operator’s domain. (M)

    -
    - -
    -
    -property codim: Integral#
    -

    Return dimension of operator’s co-domain. (N)

    -
    - -
    -
    -classmethod properties()[source]#
    -

    Mathematical properties of the operator.

    -
    -
    Return type:
    -

    Set[Property]

    -
    -
    -
    - -
    -
    -classmethod has(prop)[source]#
    -

    Verify if operator possesses supplied properties.

    -
    -
    Parameters:
    -

    prop (Property | Collection[Property]) –

    -
    -
    Return type:
    -

    bool

    -
    -
    -
    - -
    -
    -asop(cast_to)[source]#
    -

    Recast an Operator (or subclass thereof) to another Operator.

    -

    Users may call this method if the arithmetic API yields sub-optimal return types.

    -

    This method is a no-op if cast_to is a parent class of self.

    -
    -
    Parameters:
    -

    cast_to (OpC) – Target type for the recast.

    -
    -
    Returns:
    -

    op – Operator with the new interface.

    -

    Fails when cast is forbidden. -(Ex: Map -> Func if codim > 1)

    -

    -
    -
    Return type:
    -

    OpT

    -
    -
    -

    Notes

    -
      -
    • The interface of cast_to is provided via encapsulation + forwarding.

    • -
    • If self does not implement all methods from cast_to, then unimplemented methods will raise -NotImplementedError when called.

    • -
    -
    - -
    -
    -__add__(other)[source]#
    -

    Add two operators.

    -
    -
    Parameters:
    -
      -
    • self (OpT) – (A, B) Left operand.

    • -
    • other (OpT) – (C, D) Right operand.

    • -
    -
    -
    Returns:
    -

    op – Composite operator self + other

    -
    -
    Return type:
    -

    OpT

    -
    -
    -

    Notes

    -

    Operand shapes must be consistent, i.e.:

    -
    -
      -
    • have identical shape, or

    • -
    • be range-broadcastable, i.e. functional + map works.

    • -
    -
    -
    -

    See also

    -

    AddRule

    -
    -
    - -
    -
    -__sub__(other)[source]#
    -

    Subtract two operators.

    -
    -
    Parameters:
    -
      -
    • self (OpT) – (A, B) Left operand.

    • -
    • other (OpT) – (C, D) Right operand.

    • -
    -
    -
    Returns:
    -

    op – Composite operator self - other

    -
    -
    Return type:
    -

    OpT

    -
    -
    -
    - -
    -
    -__neg__()[source]#
    -

    Negate an operator.

    -
    -
    Returns:
    -

    op – Composite operator -1 * self.

    -
    -
    Return type:
    -

    OpT

    -
    -
    -
    - -
    -
    -__mul__(other)[source]#
    -

    Compose two operators, or scale an operator by a constant.

    -
    -
    Parameters:
    -
      -
    • self (OpT) – (A, B) Left operand.

    • -
    • other (Real, OpT) – (1,) scalar, or -(C, D) Right operand.

    • -
    -
    -
    Returns:
    -

    op – (A, B) scaled operator, or -(A, D) composed operator self * other.

    -
    -
    Return type:
    -

    OpT

    -
    -
    -

    Notes

    -

    If called with two operators, their shapes must be consistent, i.e. B == C.

    -
    -

    See also

    -

    ScaleRule, ChainRule

    -
    -
    - -
    -
    -__pow__(k)[source]#
    -

    Exponentiate an operator, i.e. compose it with itself.

    -
    -
    Parameters:
    -

    k (Integer) – Number of times the operator is composed with itself.

    -
    -
    Returns:
    -

    op – Exponentiated operator.

    -
    -
    Return type:
    -

    OpT

    -
    -
    -

    Notes

    -

    Exponentiation is only allowed for endomorphisms, i.e. square operators.

    -
    -

    See also

    -

    PowerRule

    -
    -
    - -
    -
    -argscale(scalar)[source]#
    -

    Scale operator’s domain.

    -
    -
    Parameters:
    -

    scalar (Real) –

    -
    -
    Returns:
    -

    op – (N, M) domain-scaled operator.

    -
    -
    Return type:
    -

    OpT

    -
    -
    -
    -

    See also

    -

    ArgScaleRule

    -
    -
    - -
    -
    -argshift(shift)[source]#
    -

    Shift operator’s domain.

    -
    -
    Parameters:
    -

    shift (Real, NDArray) – (M,) shift value

    -
    -
    Returns:
    -

    op – (N, M) domain-shifted operator.

    -
    -
    Return type:
    -

    OpT

    -
    -
    -
    -

    See also

    -

    ArgShiftRule

    -
    -
    - -
    -
    -squeeze()[source]#
    -

    Cast an Operator to the right core operator sub-type given codomain dimension.

    -
    -
    Return type:
    -

    OpT

    -
    -
    -
    - -
    -
    -expr(level=0, strip=True)[source]#
    -

    Pretty-Print the expression representation of the operator.

    -

    Useful for debugging arithmetic-induced expressions.

    -

    Example

    -
    import numpy as np
    -import pyxu.abc as pxa
    -
    -N = 5
    -op1 = pxa.LinFunc.from_array(np.arange(N))
    -op2 = pxa.LinOp.from_array(np.ones((N, N)))
    -op = ((2 * op1) + (op2 ** 3)).argshift(np.full(N, 4))
    -
    -print(op.expr())
    -# [argshift, ==> DiffMap(5, 5)
    -# .[add, ==> SquareOp(5, 5)
    -# ..[scale, ==> LinFunc(1, 5)
    -# ...LinFunc(1, 5),
    -# ...2.0],
    -# ..[exp, ==> SquareOp(5, 5)
    -# ...LinOp(5, 5),
    -# ...3]],
    -# .(5,)]
    -
    -
    -
    -
    Parameters:
    -
      -
    • level (int) –

    • -
    • strip (bool) –

    • -
    -
    -
    Return type:
    -

    str

    -
    -
    -
    - -
    - -
    -
    -class Map(shape)[source]#
    -

    Bases: Operator

    -

    Base class for real-valued maps \(\mathbf{f}: \mathbb{R}^{M} \to \mathbb{R}^{N}\).

    -

    Instances of this class must implement apply().

    -

    If \(\mathbf{f}\) is Lipschitz-continuous with known Lipschitz constant \(L\), the latter should be stored -in the lipschitz property.

    -
    -
    Parameters:
    -

    shape (tuple[Integral, Integral]) –

    -
    -
    -
    -
    -classmethod properties()[source]#
    -

    Mathematical properties of the operator.

    -
    -
    Return type:
    -

    Set[Property]

    -
    -
    -
    - -
    -
    -apply(arr)[source]#
    -

    Evaluate operator at specified point(s).

    -
    -
    Parameters:
    -

    arr (NDArray) – (…, M) input points.

    -
    -
    Returns:
    -

    out – (…, N) output points.

    -
    -
    Return type:
    -

    NDArray

    -
    -
    -
    - -
    -
    -__call__(arr)[source]#
    -

    Alias for apply().

    -
    -
    Parameters:
    -

    arr (NDArray) –

    -
    -
    Return type:
    -

    NDArray

    -
    -
    -
    - -
    -
    -property lipschitz: Real#
    -

    Return the last computed Lipschitz constant of \(\mathbf{f}\).

    -

    Notes

    -
      -
    • If a Lipschitz constant is known apriori, it can be stored in the instance as follows:

      -
      class TestOp(Map):
      -    def __init__(self, shape):
      -        super().__init__(shape=shape)
      -        self.lipschitz = 2
      -
      -op = TestOp(shape=(2, 3))
      -op.lipschitz  # => 2
      -
      -
      -

      Alternatively the Lipschitz constant can be set manually after initialization:

      -
      class TestOp(Map):
      -    def __init__(self, shape):
      -        super().__init__(shape=shape)
      -
      -op = TestOp(shape=(2,3))
      -op.lipschitz  # => inf, since unknown apriori
      -
      -op.lipschitz = 2  # post-init specification
      -op.lipschitz  # => 2
      -
      -
      -
    • -
    • lipschitz() never computes anything: -call estimate_lipschitz() manually to compute a new Lipschitz estimate:

      -
      op.lipschitz = op.estimate_lipschitz()
      -
      -
      -
    • -
    -
    - -
    -
    -estimate_lipschitz(**kwargs)[source]#
    -

    Compute a Lipschitz constant of the operator.

    -
    -
    Parameters:
    -

    kwargs (Mapping) – Class-specific kwargs to configure Lipschitz estimation.

    -
    -
    Return type:
    -

    Real

    -
    -
    -

    Notes

    -
      -
    • This method should always be callable without specifying any kwargs.

    • -
    • A constant \(L_{\mathbf{f}} > 0\) is said to be a Lipschitz constant for a map \(\mathbf{f}: -\mathbb{R}^{M} \to \mathbb{R}^{N}\) if:

      -
      -\[\|\mathbf{f}(\mathbf{x}) - \mathbf{f}(\mathbf{y})\|_{\mathbb{R}^{N}} -\leq -L_{\mathbf{f}} \|\mathbf{x} - \mathbf{y}\|_{\mathbb{R}^{M}}, -\qquad -\forall \mathbf{x}, \mathbf{y}\in \mathbb{R}^{M},\]
      -

      where \(\|\cdot\|_{\mathbb{R}^{M}}\) and \(\|\cdot\|_{\mathbb{R}^{N}}\) are the canonical norms on -their respective spaces.

      -

      The smallest Lipschitz constant of a map is called the optimal Lipschitz constant.

      -
    • -
    -
    - -
    - -
    -
    -class Func(shape)[source]#
    -

    Bases: Map

    -

    Base class for real-valued functionals \(f: \mathbb{R}^{M} \to \mathbb{R}\cup\{+\infty\}\).

    -

    Instances of this class must implement apply().

    -

    If \(f\) is Lipschitz-continuous with known Lipschitz constant \(L\), the latter should be stored in the -lipschitz property.

    -
    -
    Parameters:
    -

    shape (tuple[Integral, Integral]) –

    -
    -
    -
    -
    -asloss(data=None)[source]#
    -

    Transform a functional into a loss functional.

    -

    The meaning of asloss() is operator-dependent: always examine the output of -expr() to determine the loss-notion involved.

    -
    -
    Parameters:
    -

    data (NDArray) – (M,) input.

    -
    -
    Returns:
    -

    op – (1, M) loss function. -If data is unspecified, returns self.

    -
    -
    Return type:
    -

    OpT

    -
    -
    -
    - -
    - -
    -
    -class DiffMap(shape)[source]#
    -

    Bases: Map

    -

    Base class for real-valued differentiable maps \(\mathbf{f}: \mathbb{R}^{M} \to \mathbb{R}^{N}\).

    -

    Instances of this class must implement apply() and jacobian().

    -

    If \(\mathbf{f}\) is Lipschitz-continuous with known Lipschitz constant \(L\), the latter should be stored -in the lipschitz property.

    -

    If \(\mathbf{J}_{\mathbf{f}}\) is Lipschitz-continuous with known Lipschitz constant \(\partial L\), the -latter should be stored in the diff_lipschitz property.

    -
    -
    Parameters:
    -

    shape (tuple[Integral, Integral]) –

    -
    -
    -
    -
    -jacobian(arr)[source]#
    -

    Evaluate the Jacobian of \(\mathbf{f}\) at the specified point.

    -
    -
    Parameters:
    -

    arr (NDArray) – (M,) evaluation point.

    -
    -
    Returns:
    -

    op – (N, M) Jacobian operator at point arr.

    -
    -
    Return type:
    -

    OpT

    -
    -
    -

    Notes

    -

    Let \(\mathbf{f}=[f_{1}, \ldots, f_{N}]: \mathbb{R}^{M} \to \mathbb{R}^{N}\) be a differentiable -multi-dimensional map. The Jacobian (or differential) of \(\mathbf{f}\) at \(\mathbf{z} \in -\mathbb{R}^{M}\) is defined as the best linear approximator of \(\mathbf{f}\) near \(\mathbf{z}\), in the -following sense:

    -
    -\[\mathbf{f}(\mathbf{x}) - \mathbf{f}(\mathbf{z}) -= -\mathbf{J}_{\mathbf{f}}(\mathbf{z}) (\mathbf{x} - \mathbf{z}) -+ -o(\| \mathbf{x} - \mathbf{z} \|) -\quad -\text{as} \quad \mathbf{x} \to \mathbf{z}.\]
    -

    The Jacobian admits the following matrix representation:

    -
    -\[[\mathbf{J}_{\mathbf{f}}(\mathbf{x})]_{ij} -:= -\frac{\partial f_{i}}{\partial x_{j}}(\mathbf{x}), -\qquad -\forall (i,j) \in \{1,\ldots,N\} \times \{1,\ldots,M\}.\]
    -
    - -
    -
    -property diff_lipschitz: Real#
    -

    Return the last computed Lipschitz constant of \(\mathbf{J}_{\mathbf{f}}\).

    -

    Notes

    -
      -
    • If a diff-Lipschitz constant is known apriori, it can be stored in the instance as follows:

      -
      class TestOp(DiffMap):
      -    def __init__(self, shape):
      -        super().__init__(shape=shape)
      -        self.diff_lipschitz = 2
      -
      -op = TestOp(shape=(2, 3))
      -op.diff_lipschitz  # => 2
      -
      -
      -

      Alternatively the diff-Lipschitz constant can be set manually after initialization:

      -
      class TestOp(DiffMap):
      -    def __init__(self, shape):
      -        super().__init__(shape=shape)
      -
      -op = TestOp(shape=(2,3))
      -op.diff_lipschitz  # => inf, since unknown apriori
      -
      -op.diff_lipschitz = 2  # post-init specification
      -op.diff_lipschitz  # => 2
      -
      -
      -
    • -
    • diff_lipschitz() never computes anything: -call estimate_diff_lipschitz() manually to compute a new diff-Lipschitz estimate:

      -
      op.diff_lipschitz = op.estimate_diff_lipschitz()
      -
      -
      -
    • -
    -
    - -
    -
    -estimate_diff_lipschitz(**kwargs)[source]#
    -

    Compute a Lipschitz constant of jacobian().

    -
    -
    Parameters:
    -

    kwargs (Mapping) – Class-specific kwargs to configure diff-Lipschitz estimation.

    -
    -
    Return type:
    -

    Real

    -
    -
    -

    Notes

    -
      -
    • This method should always be callable without specifying any kwargs.

    • -
    • A Lipschitz constant \(L_{\mathbf{J}_{\mathbf{f}}} > 0\) of the Jacobian map -\(\mathbf{J}_{\mathbf{f}}: \mathbb{R}^{M} \to \mathbb{R}^{N \times M}\) is such that:

      -
      -\[\|\mathbf{J}_{\mathbf{f}}(\mathbf{x}) - \mathbf{J}_{\mathbf{f}}(\mathbf{y})\|_{\mathbb{R}^{N \times M}} -\leq -L_{\mathbf{J}_{\mathbf{f}}} \|\mathbf{x} - \mathbf{y}\|_{\mathbb{R}^{M}}, -\qquad -\forall \mathbf{x}, \mathbf{y} \in \mathbb{R}^{M},\]
      -

      where \(\|\cdot\|_{\mathbb{R}^{N \times M}}\) and \(\|\cdot\|_{\mathbb{R}^{M}}\) are the canonical -norms on their respective spaces.

      -

      The smallest Lipschitz constant of the Jacobian is called the optimal diff-Lipschitz constant.

      -
    • -
    -
    - -
    - -
    -
    -class DiffFunc(shape)[source]#
    -

    Bases: DiffMap, Func

    -

    Base class for real-valued differentiable functionals \(f: \mathbb{R}^{M} \to \mathbb{R}\).

    -

    Instances of this class must implement apply() and grad().

    -

    If \(f\) and/or its derivative \(f'\) are Lipschitz-continuous with known Lipschitz constants \(L\) and -\(\partial L\), the latter should be stored in the lipschitz and -diff_lipschitz properties.

    -
    -
    Parameters:
    -

    shape (tuple[Integral, Integral]) –

    -
    -
    -
    -
    -grad(arr)[source]#
    -

    Evaluate operator gradient at specified point(s).

    -
    -
    Parameters:
    -

    arr (NDArray) – (…, M) input points.

    -
    -
    Returns:
    -

    out – (…, M) gradients.

    -
    -
    Return type:
    -

    NDArray

    -
    -
    -

    Notes

    -

    The gradient of a functional \(f: \mathbb{R}^{M} \to \mathbb{R}\) is given, for every \(\mathbf{x} \in -\mathbb{R}^{M}\), by

    -
    -\[\begin{split}\nabla f(\mathbf{x}) -:= -\left[\begin{array}{c} -\frac{\partial f}{\partial x_{1}}(\mathbf{x}) \\ -\vdots \\ -\frac{\partial f}{\partial x_{M}}(\mathbf{x}) -\end{array}\right].\end{split}\]
    -
    - -
    - -
    -
    -class ProxFunc(shape)[source]#
    -

    Bases: Func

    -

    Base class for real-valued proximable functionals \(f: \mathbb{R}^{M} \to \mathbb{R} \cup \{+\infty\}\).

    -

    A functional \(f: \mathbb{R}^{M} \to \mathbb{R} \cup \{+\infty\}\) is said proximable if its proximity -operator (see prox() for a definition) admits a simple closed-form expression or -can be evaluated efficiently and with high accuracy.

    -

    Instances of this class must implement apply() and prox().

    -

    If \(f\) is Lipschitz-continuous with known Lipschitz constant \(L\), the latter should be stored in the -lipschitz property.

    -
    -
    Parameters:
    -

    shape (tuple[Integral, Integral]) –

    -
    -
    -
    -
    -prox(arr, tau)[source]#
    -

    Evaluate proximity operator of \(\tau f\) at specified point(s).

    -
    -
    Parameters:
    -
      -
    • arr (NDArray) – (…, M) input points.

    • -
    • tau (Real) – Positive scale factor.

    • -
    -
    -
    Returns:
    -

    out – (…, M) proximal evaluations.

    -
    -
    Return type:
    -

    NDArray

    -
    -
    -

    Notes

    -

    For \(\tau >0\), the proximity operator of a scaled functional \(f: \mathbb{R}^{M} \to \mathbb{R}\) is -defined as:

    -
    -\[\mathbf{\text{prox}}_{\tau f}(\mathbf{z}) -:= -\arg\min_{\mathbf{x}\in\mathbb{R}^{M}} f(x)+\frac{1}{2\tau} \|\mathbf{x}-\mathbf{z}\|_{2}^{2}, -\quad -\forall \mathbf{z} \in \mathbb{R}^{M}.\]
    -
    - -
    -
    -fenchel_prox(arr, sigma)[source]#
    -

    Evaluate proximity operator of \(\sigma f^{\ast}\), the scaled Fenchel conjugate of \(f\), at specified -point(s).

    -
    -
    Parameters:
    -
      -
    • arr (NDArray) – (…, M) input points.

    • -
    • sigma (Real) – Positive scale factor.

    • -
    -
    -
    Returns:
    -

    out – (…, M) proximal evaluations.

    -
    -
    Return type:
    -

    NDArray

    -
    -
    -

    Notes

    -

    For \(\sigma > 0\), the Fenchel conjugate is defined as:

    -
    -\[f^{\ast}(\mathbf{z}) -:= -\max_{\mathbf{x}\in\mathbb{R}^{M}} \langle \mathbf{x},\mathbf{z} \rangle - f(\mathbf{x}).\]
    -

    From Moreau’s identity, its proximal operator is given by:

    -
    -\[\mathbf{\text{prox}}_{\sigma f^{\ast}}(\mathbf{z}) -= -\mathbf{z} - \sigma \mathbf{\text{prox}}_{f/\sigma}(\mathbf{z}/\sigma).\]
    -
    - -
    -
    -moreau_envelope(mu)[source]#
    -

    Approximate proximable functional \(f\) by its Moreau envelope \(f^{\mu}\).

    -
    -
    Parameters:
    -

    mu (Real) – Positive regularization parameter.

    -
    -
    Returns:
    -

    op – Differential Moreau envelope.

    -
    -
    Return type:
    -

    OpT

    -
    -
    -

    Notes

    -

    Consider a convex non-smooth proximable functional \(f: \mathbb{R}^{M} \to \mathbb{R} \cup \{+\infty\}\) and -a regularization parameter \(\mu > 0\). The \(\mu\)-Moreau envelope (or Moreau-Yoshida envelope) of -\(f\) is given by

    -
    -\[f^{\mu}(\mathbf{x}) -= -\min_{\mathbf{z} \in \mathbb{R}^{M}} -f(\mathbf{z}) -\quad + \quad -\frac{1}{2\mu} \|\mathbf{x} - \mathbf{z}\|^{2}.\]
    -

    The parameter \(\mu\) controls the trade-off between the regularity properties of \(f^{\mu}\) and the -approximation error incurred by the Moreau-Yoshida regularization.

    -

    The Moreau envelope inherits the convexity of \(f\) and is gradient-Lipschitz (with Lipschitz constant -\(\mu^{-1}\)), even if \(f\) is non-smooth. Its gradient is moreover given by:

    -
    -\[\nabla f^{\mu}(\mathbf{x}) -= -\mu^{-1} \left(\mathbf{x} - \text{prox}_{\mu f}(\mathbf{x})\right).\]
    -

    In addition, \(f^{\mu}\) envelopes \(f\) from below: \(f^{\mu}(\mathbf{x}) \leq f(\mathbf{x})\). -This envelope becomes tighter as \(\mu \to 0\):

    -
    -\[\lim_{\mu \to 0} f^{\mu}(\mathbf{x}) = f(\mathbf{x}).\]
    -

    Finally, it can be shown that the minimizers of \(f\) and \(f^{\mu}\) coincide, and that the Fenchel -conjugate of \(f^{\mu}\) is strongly-convex.

    -

    Example

    -

    Construct and plot the Moreau envelope of the \(\ell_{1}\)-norm:

    -
    import numpy as np
    -import matplotlib.pyplot as plt
    -from pyxu.abc import ProxFunc
    -
    -class L1Norm(ProxFunc):
    -    def __init__(self, dim: int):
    -        super().__init__(shape=(1, dim))
    -    def apply(self, arr):
    -        return np.linalg.norm(arr, axis=-1, keepdims=True, ord=1)
    -    def prox(self, arr, tau):
    -        return np.clip(np.abs(arr)-tau, a_min=0, a_max=None) * np.sign(arr)
    -
    -N = 512
    -l1_norm = L1Norm(dim=N)
    -mus = [0.1, 0.5, 1]
    -smooth_l1_norms = [l1_norm.moreau_envelope(mu) for mu in mus]
    -
    -x = np.linspace(-1, 1, N)[:, None]
    -labels=['mu=0']
    -labels.extend([f'mu={mu}' for mu in mus])
    -plt.figure()
    -plt.plot(x, l1_norm(x))
    -for f in smooth_l1_norms:
    -    plt.plot(x, f(x))
    -plt.legend(labels)
    -plt.title('Moreau Envelope')
    -
    -labels=[f'mu={mu}' for mu in mus]
    -plt.figure()
    -for f in smooth_l1_norms:
    -    plt.plot(x, f.grad(x))
    -plt.legend(labels)
    -plt.title('Derivative of Moreau Envelope')
    -
    -
    -

    (Source code)

    -
    -../../_images/operator-1_00.png -
    -

    (png, hires.png, pdf)#

    -
    -
    -
    -../../_images/operator-1_01.png -
    -

    (png, hires.png, pdf)#

    -
    -
    -
    - -
    - -
    -
    -class ProxDiffFunc(shape)[source]#
    -

    Bases: ProxFunc, DiffFunc

    -

    Base class for real-valued differentiable and proximable functionals \(f:\mathbb{R}^{M} \to \mathbb{R}\).

    -

    Instances of this class must implement apply(), grad(), and -prox().

    -

    If \(f\) and/or its derivative \(f'\) are Lipschitz-continuous with known Lipschitz constants \(L\) and -\(\partial L\), the latter should be stored in the lipschitz and -diff_lipschitz properties.

    -
    -
    Parameters:
    -

    shape (tuple[Integral, Integral]) –

    -
    -
    -
    - -
    -
    -class QuadraticFunc(shape, Q=None, c=None, t=0)[source]#
    -

    Bases: ProxDiffFunc

    -

    Base class for quadratic functionals \(f: \mathbb{R}^{M} \to \mathbb{R} \cup \{+\infty\}\).

    -

    The quadratic functional is defined as:

    -
    -\[f(\mathbf{x}) -= -\frac{1}{2} \langle\mathbf{x}, \mathbf{Q}\mathbf{x}\rangle -+ -\mathbf{c}^T\mathbf{x} -+ -t, -\qquad \forall \mathbf{x} \in \mathbb{R}^{M},\]
    -

    where \(Q\) is a positive-definite operator \(\mathbf{Q}:\mathbb{R}^{M} \to \mathbb{R}^{M}\), -\(\mathbf{c} \in \mathbb{R}^{M}\), and \(t > 0\).

    -

    Its gradient is given by:

    -
    -\[\nabla f(\mathbf{x}) = \mathbf{Q}\mathbf{x} + \mathbf{c}.\]
    -

    Its proximity operator by:

    -
    -\[\text{prox}_{\tau f}(x) -= -\left( - \mathbf{Q} + \tau^{-1} \mathbf{Id} -\right)^{-1} -\left( - \tau^{-1}\mathbf{x} - \mathbf{c} -\right).\]
    -

    In practice the proximity operator is evaluated via CG.

    -

    The Lipschitz constant \(L\) of a quadratic on an unbounded domain is unbounded. The Lipschitz constant -\(\partial L\) of \(\nabla f\) is given by the spectral norm of \(\mathbf{Q}\).

    -
    -
    Parameters:
    -
    -
    -
    -
    -
    -__init__(shape, Q=None, c=None, t=0)[source]#
    -
    -
    Parameters:
    -
      -
    • Q (PosDefOp) – (N, N) positive-definite operator. (Default: Identity)

    • -
    • c (LinFunc) – (1, N) linear functional. (Default: NullFunc)

    • -
    • t (Real) – offset. (Default: 0)

    • -
    • shape (tuple[Integral, Integral]) –

    • -
    -
    -
    -
    - -
    -
    -_quad_spec()[source]#
    -

    Canonical quadratic parameterization.

    -

    Useful for some internal methods, and overloaded during operator arithmetic.

    -
    - -
    - -
    -
    -class LinOp(shape)[source]#
    -

    Bases: DiffMap

    -

    Base class for real-valued linear operators \(\mathbf{A}: \mathbb{R}^{M} \to \mathbb{R}^{N}\).

    -

    Instances of this class must implement apply() and adjoint().

    -

    If known, the Lipschitz constant \(L\) should be stored in the lipschitz property.

    -

    The Jacobian of a linear map \(\mathbf{A}\) is constant.

    -
    -
    Parameters:
    -

    shape (tuple[Integral, Integral]) –

    -
    -
    -
    -
    -adjoint(arr)[source]#
    -

    Evaluate operator adjoint at specified point(s).

    -
    -
    Parameters:
    -

    arr (NDArray) – (…, N) input points.

    -
    -
    Returns:
    -

    out – (…, M) adjoint evaluations.

    -
    -
    Return type:
    -

    NDArray

    -
    -
    -

    Notes

    -

    The adjoint \(\mathbf{A}^{\ast}: \mathbb{R}^{N} \to \mathbb{R}^{M}\) of \(\mathbf{A}: \mathbb{R}^{M} -\to \mathbb{R}^{N}\) is defined as:

    -
    -\[\langle \mathbf{x}, \mathbf{A}^{\ast}\mathbf{y}\rangle_{\mathbb{R}^{M}} -:= -\langle \mathbf{A}\mathbf{x}, \mathbf{y}\rangle_{\mathbb{R}^{N}}, -\qquad -\forall (\mathbf{x},\mathbf{y})\in \mathbb{R}^{M} \times \mathbb{R}^{N}.\]
    -
    - -
    -
    -property T: OpT#
    -

    Return the (M, N) adjoint of \(\mathbf{A}\).

    -
    - -
    -
    -to_sciop(dtype=None, gpu=False)[source]#
    -

    Cast a LinOp to a CPU/GPU LinearOperator, compatible with -the matrix-free linear algebra routines of scipy.sparse.linalg.

    -
    -
    Parameters:
    -
      -
    • dtype (DType) – Working precision of the linear operator.

    • -
    • gpu (bool) – Operate on CuPy inputs (True) vs. NumPy inputs (False).

    • -
    -
    -
    Returns:
    -

    op – Linear operator object compliant with SciPy’s interface.

    -
    -
    Return type:
    -

    LinearOperator

    -
    -
    -
    - -
    -
    -estimate_lipschitz(**kwargs)[source]#
    -

    Compute a Lipschitz constant of the operator.

    -
    -
    Parameters:
    -
      -
    • method ("svd" | "trace") –

        -
      • If svd, compute the optimal Lipschitz constant.

      • -
      • If trace, compute an upper bound. (Default)

      • -
      -

    • -
    • kwargs

      Optional kwargs passed on to:

      - -

    • -
    -
    -
    Return type:
    -

    Real

    -
    -
    -

    Notes

    -
      -
    • The tightest Lipschitz constant is given by the spectral norm of the operator \(\mathbf{A}\): -\(\|\mathbf{A}\|_{2}\). It can be computed via the SVD, which is compute-intensive task for large -operators. In this setting, it may be advantageous to overestimate the Lipschitz constant with the Frobenius -norm of \(\mathbf{A}\) since \(\|\mathbf{A}\|_{F} \geq \|\mathbf{A}\|_{2}\).

      -

      \(\|\mathbf{A}\|_{F}\) can be efficiently approximated by computing the trace of \(\mathbf{A}^{\ast} -\mathbf{A}\) (or \(\mathbf{A}\mathbf{A}^{\ast}\)) via the Hutch++ stochastic algorithm.

      -
    • -
    • \(\|\mathbf{A}\|_{F}\) is upper-bounded by \(\|\mathbf{A}\|_{F} \leq \sqrt{n} \|\mathbf{A}\|_{2}\), -where the equality is reached (worst-case scenario) when the eigenspectrum of the linear operator is flat.

    • -
    -
    - -
    -
    -svdvals(k, which='LM', gpu=False, **kwargs)[source]#
    -

    Compute singular values of a linear operator.

    -
    -
    Parameters:
    -
      -
    • k (Integer) – Number of singular values to compute.

    • -
    • which ("LM" | "SM") –

      Which k singular values to find:

      -
      -
        -
      • LM: largest magnitude

      • -
      • SM: smallest magnitude

      • -
      -
      -

    • -
    • gpu (bool) – If True the singular value decomposition is performed on the GPU.

    • -
    • kwargs (Mapping) – Additional kwargs accepted by svds().

    • -
    -
    -
    Returns:
    -

    D – (k,) singular values in ascending order.

    -
    -
    Return type:
    -

    NDArray

    -
    -
    -
    - -
    -
    -asarray(xp=None, dtype=None)[source]#
    -

    Matrix representation of the linear operator.

    -
    -
    Parameters:
    -
      -
    • xp (ArrayModule) – Which array module to use to represent the output. (Default: NumPy.)

    • -
    • dtype (DType) – Precision of the array. (Default: current runtime precision.)

    • -
    -
    -
    Returns:
    -

    A – (N, M) array-representation of the operator.

    -
    -
    Return type:
    -

    NDArray

    -
    -
    -
    -

    Note

    -

    This generic implementation assumes the operator is backend-agnostic. Thus, when defining a new -backend-specific operator, asarray() may need to be overriden.

    -
    -
    - -
    -
    -gram()[source]#
    -

    Gram operator \(\mathbf{A}^{\ast} \mathbf{A}: \mathbb{R}^{M} \to \mathbb{R}^{M}\).

    -
    -
    Returns:
    -

    op – (M, M) Gram operator.

    -
    -
    Return type:
    -

    OpT

    -
    -
    -
    -

    Note

    -

    By default the Gram is computed by the composition self.T * self. This may not be the fastest way to -compute the Gram operator. If the Gram can be computed more efficiently (e.g. with a convolution), the user -should re-define this method.

    -
    -
    - -
    -
    -cogram()[source]#
    -

    Co-Gram operator \(\mathbf{A}\mathbf{A}^{\ast}:\mathbb{R}^{N} \to \mathbb{R}^{N}\).

    -
    -
    Returns:
    -

    op – (N, N) Co-Gram operator.

    -
    -
    Return type:
    -

    OpT

    -
    -
    -
    -

    Note

    -

    By default the co-Gram is computed by the composition self * self.T. This may not be the fastest way to -compute the co-Gram operator. If the co-Gram can be computed more efficiently (e.g. with a convolution), the -user should re-define this method.

    -
    -
    - -
    -
    -pinv(arr, damp, kwargs_init=None, kwargs_fit=None)[source]#
    -

    Evaluate the Moore-Penrose pseudo-inverse \(\mathbf{A}^{\dagger}\) at specified point(s).

    -
    -
    Parameters:
    -
      -
    • arr (NDArray) – (…, N) input points.

    • -
    • damp (Real) – Positive dampening factor regularizing the pseudo-inverse.

    • -
    • kwargs_init (Mapping) – Optional kwargs to be passed to CG()’s __init__() method.

    • -
    • kwargs_fit (Mapping) – Optional kwargs to be passed to CG()’s fit() method.

    • -
    -
    -
    Returns:
    -

    out – (…, M) pseudo-inverse(s).

    -
    -
    Return type:
    -

    NDArray

    -
    -
    -

    Notes

    -

    The Moore-Penrose pseudo-inverse of an operator \(\mathbf{A}: \mathbb{R}^{M} \to \mathbb{R}^{N}\) is defined -as the operator \(\mathbf{A}^{\dagger}: \mathbb{R}^{N} \to \mathbb{R}^{M}\) verifying the Moore-Penrose -conditions:

    -
    -
      -
    1. \(\mathbf{A} \mathbf{A}^{\dagger} \mathbf{A} = \mathbf{A}\),

    2. -
    3. \(\mathbf{A}^{\dagger} \mathbf{A} \mathbf{A}^{\dagger} = \mathbf{A}^{\dagger}\),

    4. -
    5. \((\mathbf{A}^{\dagger} \mathbf{A})^{\ast} = \mathbf{A}^{\dagger} \mathbf{A}\),

    6. -
    7. \((\mathbf{A} \mathbf{A}^{\dagger})^{\ast} = \mathbf{A} \mathbf{A}^{\dagger}\).

    8. -
    -
    -

    This operator exists and is unique for any finite-dimensional linear operator. The action of the pseudo-inverse -\(\mathbf{A}^{\dagger} \mathbf{y}\) for every \(\mathbf{y} \in \mathbb{R}^{N}\) can be computed in -matrix-free fashion by solving the normal equations:

    -
    -\[\mathbf{A}^{\ast} \mathbf{A} \mathbf{x} = \mathbf{A}^{\ast} \mathbf{y} -\quad\Leftrightarrow\quad -\mathbf{x} = \mathbf{A}^{\dagger} \mathbf{y}, -\quad -\forall (\mathbf{x}, \mathbf{y}) \in \mathbb{R}^{M} \times \mathbb{R}^{N}.\]
    -

    In the case of severe ill-conditioning, it is possible to consider the dampened normal equations for a -numerically-stabler approximation of \(\mathbf{A}^{\dagger} \mathbf{y}\):

    -
    -\[(\mathbf{A}^{\ast} \mathbf{A} + \tau I) \mathbf{x} = \mathbf{A}^{\ast} \mathbf{y},\]
    -

    where \(\tau > 0\) corresponds to the damp parameter.

    -
    - -
    -
    -dagger(damp, kwargs_init=None, kwargs_fit=None)[source]#
    -

    Return the Moore-Penrose pseudo-inverse operator \(\mathbf{A}^\dagger\).

    -
    -
    Parameters:
    -
      -
    • damp (Real) – Positive dampening factor regularizing the pseudo-inverse.

    • -
    • kwargs_init (Mapping) – Optional kwargs to be passed to CG()’s __init__() method.

    • -
    • kwargs_fit (Mapping) – Optional kwargs to be passed to CG()’s fit() method.

    • -
    -
    -
    Returns:
    -

    op – (M, N) Moore-Penrose pseudo-inverse operator.

    -
    -
    Return type:
    -

    OpT

    -
    -
    -
    - -
    -
    -classmethod from_array(A, enable_warnings=True)[source]#
    -

    Instantiate a LinOp from its array representation.

    -
    -
    Parameters:
    -
      -
    • A (NDArray) – (N, M) array

    • -
    • enable_warnings (bool) – If True, emit a warning in case of precision mis-match issues.

    • -
    -
    -
    Returns:
    -

    op – (N, M) linear operator

    -
    -
    Return type:
    -

    OpT

    -
    -
    -
    - -
    - -
    -
    -class SquareOp(shape)[source]#
    -

    Bases: LinOp

    -

    Base class for square linear operators, i.e. \(\mathbf{A}: \mathbb{R}^{M} \to \mathbb{R}^{M}\) -(endomorphsisms).

    -
    -
    Parameters:
    -

    shape (tuple[Integral, Integral]) –

    -
    -
    -
    -
    -trace(**kwargs)[source]#
    -

    Compute trace of the operator.

    -
    -
    Parameters:
    -
      -
    • method ("explicit" | "hutchpp") –

        -
      • If explicit, compute the exact trace.

      • -
      • If hutchpp, compute an approximation.

      • -
      -

    • -
    • kwargs (Mapping) –

      Optional kwargs passed to:

      - -

    • -
    -
    -
    Returns:
    -

    tr – Trace estimate.

    -
    -
    Return type:
    -

    Real

    -
    -
    -
    - -
    - -
    -
    -class NormalOp(shape)[source]#
    -

    Bases: SquareOp

    -

    Base class for normal operators.

    -

    Normal operators satisfy the relation \(\mathbf{A} \mathbf{A}^{\ast} = \mathbf{A}^{\ast} \mathbf{A}\). It can -be shown that an operator is normal iff it is -unitarily diagonalizable, i.e. \(\mathbf{A} = \mathbf{U} \mathbf{D} \mathbf{U}^{\ast}\).

    -
    -
    Parameters:
    -

    shape (tuple[Integral, Integral]) –

    -
    -
    -
    - -
    -
    -class SelfAdjointOp(shape)[source]#
    -

    Bases: NormalOp

    -

    Base class for self-adjoint operators.

    -

    Self-adjoint operators satisfy the relation \(\mathbf{A}^{\ast} = \mathbf{A}\).

    -
    -
    Parameters:
    -

    shape (tuple[Integral, Integral]) –

    -
    -
    -
    - -
    -
    -class UnitOp(shape)[source]#
    -

    Bases: NormalOp

    -

    Base class for unitary operators.

    -

    Unitary operators satisfy the relation \(\mathbf{A} \mathbf{A}^{\ast} = \mathbf{A}^{\ast} \mathbf{A} = I\).

    -
    -
    Parameters:
    -

    shape (tuple[Integral, Integral]) –

    -
    -
    -
    - -
    -
    -class ProjOp(shape)[source]#
    -

    Bases: SquareOp

    -

    Base class for projection operators.

    -

    Projection operators are idempotent, i.e. \(\mathbf{A}^{2} = \mathbf{A}\).

    -
    -
    Parameters:
    -

    shape (tuple[Integral, Integral]) –

    -
    -
    -
    - -
    -
    -class OrthProjOp(shape)[source]#
    -

    Bases: ProjOp, SelfAdjointOp

    -

    Base class for orthogonal projection operators.

    -

    Orthogonal projection operators are idempotent and self-adjoint, i.e. \(\mathbf{A}^{2} = \mathbf{A}\) and -\(\mathbf{A}^{\ast} = \mathbf{A}\).

    -
    -
    Parameters:
    -

    shape (tuple[Integral, Integral]) –

    -
    -
    -
    - -
    -
    -class PosDefOp(shape)[source]#
    -

    Bases: SelfAdjointOp

    -

    Base class for positive-definite operators.

    -
    -
    Parameters:
    -

    shape (tuple[Integral, Integral]) –

    -
    -
    -
    - -
    -
    -class LinFunc(shape)[source]#
    -

    Bases: ProxDiffFunc, LinOp

    -

    Base class for real-valued linear functionals \(f: \mathbb{R}^{M} \to \mathbb{R}\).

    -

    Instances of this class must implement apply(), and adjoint().

    -

    If known, the Lipschitz constant \(L\) should be stored in the lipschitz property.

    -
    -
    Parameters:
    -

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    pyxu.abc.solver#

    -
    -
    -class Solver(*, folder=None, exist_ok=False, stop_rate=1, writeback_rate=None, verbosity=None, show_progress=True, log_var=frozenset({}))[source]#
    -

    Bases: object

    -

    Iterative solver for minimization problems of the form \(\hat{x} = \arg\min_{x \in \mathbb{R}^{N}} -\mathcal{F}(x)\), where the form of \(\mathcal{F}\) is solver-dependent.

    -

    Solver provides a versatile API for solving optimisation problems, with the following features:

    -
      -
    • manual/automatic/background execution of solver iterations via parameters provided to -fit(). (See below.)

    • -
    • automatic checkpointing of solver progress, providing a safe restore point in case of faulty numerical code. Each -solver instance backs its state and final output to a folder on disk for post-analysis. In particular -fit() will never crash: detailed exception information will always be available in a -logfile for post-analysis.

    • -
    • arbitrary specification of complex stopping criteria via the StoppingCriterion class.

    • -
    • solve for multiple initial points in parallel.

    • -
    -

    To implement a new iterative solver, users need to sub-class Solver and overwrite the methods -below:

    - -

    Advanced functionalities of Solver are automatically inherited by sub-classes.

    -

    Examples

    -

    Here are examples on how to solve minimization problems with this class:

    -
    slvr = Solver()
    -
    -### 1. Blocking mode: .fit() does not return until solver has stopped.
    ->>> slvr.fit(mode=SolverMode.BLOCK, ...)
    ->>> data, hist = slvr.stats()  # final output of solver.
    -
    -### 2. Async mode: solver iterations run in the background.
    ->>> slvr.fit(mode=SolverMode.ASYNC, ...)
    ->>> print('test')  # you can do something in between.
    ->>> slvr.busy()  # or check whether the solver already stopped.
    ->>> slvr.stop()  # and preemptively force it to stop.
    ->>> data, hist = slvr.stats()  # then query the result after a (potential) force-stop.
    -
    -### 3. Manual mode: fine-grain control of solver data per iteration.
    ->>> slvr.fit(mode=SolverMode.MANUAL, ...)
    ->>> for data in slvr.steps():
    -...     # Do something with the logged variables after each iteration.
    -...     pass  # solver has stopped after the loop.
    ->>> data, hist = slvr.stats()  # final output of solver.
    -
    -
    -
    -
    Parameters:
    -
    -
    -
    -
    -
    -_mstate: dict[str, Any]#
    -

    Mathematical state.

    -
    - -
    -
    -_astate: dict[str, Any]#
    -

    Book-keeping (non-math) state.

    -
    - -
    -
    -__init__(*, folder=None, exist_ok=False, stop_rate=1, writeback_rate=None, verbosity=None, show_progress=True, log_var=frozenset({}))[source]#
    -
    -
    Parameters:
    -
      -
    • folder (Path) – Directory on disk where instance data should be stored. A location will be automatically chosen if -unspecified. (Default: OS-dependent tempdir.)

    • -
    • exist_ok (bool) – If folder is specified and exist_ok is false (default), FileExistsError is raised if the -target directory already exists.

    • -
    • stop_rate (Integer) – Rate at which solver evaluates stopping criteria.

    • -
    • writeback_rate (Integer) – Rate at which solver checkpoints are written to disk. No checkpointing is done if unspecified: only the -final solver output will be written back to disk. Must be a multiple of stop_rate if specified.

    • -
    • verbosity (Integer) – Rate at which stopping criteria statistics are logged. Must be a multiple of stop_rate. Defaults to -stop_rate if unspecified.

    • -
    • show_progress (bool) – If True (default) and fit() is run with mode=BLOCK, then statistics are also -logged to stdout.

    • -
    • log_var (VarName) – Variables from the solver’s math-state (_mstate) to be logged per iteration. -These are the variables made available when calling stats().

    • -
    -
    -
    -

    Notes

    -
      -
    • Partial device<>CPU synchronization takes place when stopping-criteria are evaluated. Increasing stop_rate -is advised to reduce the effect of such transfers when applicable.

    • -
    • Full device<>CPU synchronization takes place at checkpoint-time. Increasing writeback_rate is advised to -reduce the effect of such transfers when applicable.

    • -
    -
    - -
    -
    -fit(**kwargs)[source]#
    -

    Solve minimization problem(s) defined in __init__(), with the provided run-specifc -parameters.

    -
    -
    Parameters:
    -
      -
    • kwargs – See class-level docstring for class-specific keyword parameters.

    • -
    • stop_crit (StoppingCriterion) – Stopping criterion to end solver iterations. If unspecified, defaults to -default_stop_crit().

    • -
    • mode (SolverMode) –

      Execution mode. -See Solver for usage examples.

      -

      Useful method pairs depending on the execution mode:

      - -

    • -
    • track_objective (bool) – Auto-compute objective function every time stopping criterion is evaluated.

    • -
    -
    -
    -
    - -
    -
    -m_init(**kwargs)[source]#
    -

    Set solver’s initial mathematical state based on kwargs provided to fit().

    -

    This method must only manipulate _mstate.

    -

    After calling this method, the solver must be able to complete its 1st iteration via a call to -m_step().

    -
    - -
    -
    -m_step()[source]#
    -

    Perform one (mathematical) step.

    -

    This method must only manipulate _mstate.

    -
    - -
    -
    -steps(n=None)[source]#
    -

    Generator of logged variables after each iteration.

    -

    The i-th call to next() on this object returns the logged variables after the i-th solver iteration.

    -

    This method is only usable after calling fit() with mode=MANUAL. See -Solver for usage examples.

    -

    There is no guarantee that a checkpoint on disk exists when the generator is exhausted. (Reason: potential -exceptions raised during solver’s progress.) Users should invoke writeback() -afterwards if needed.

    -
    -
    Parameters:
    -

    n (Integer) –

    Maximum number of next() calls allowed before exhausting the generator. Defaults to infinity if -unspecified.

    -

    The generator will terminate prematurely if the solver naturally stops before n calls to next() -are made.

    -

    -
    -
    Return type:
    -

    Generator

    -
    -
    -
    - -
    -
    -stats()[source]#
    -

    Query solver state.

    -
    -
    Returns:
    -

      -
    • data (Mapping) – Value(s) of log_var (s) after last iteration.

    • -
    • history (numpy.ndarray, None) – (N_iter,) records of stopping-criteria values per iteration.

    • -
    -

    -
    -
    Return type:
    -

    tuple[dict[str, Real | NDArray | None], ndarray | None]

    -
    -
    -

    Notes

    -

    If any of the log_var (s) and/or history are not (yet) known at query time, None is returned.

    -
    - -
    -
    -property workdir: str | Path#
    -
    -
    Returns:
    -

    wd – Absolute path to the directory on disk where instance data is stored.

    -
    -
    Return type:
    -

    Path

    -
    -
    -
    - -
    -
    -property logfile: str | Path#
    -
    -
    Returns:
    -

    lf – Absolute path to the log file on disk where stopping criteria statistics are logged.

    -
    -
    Return type:
    -

    Path

    -
    -
    -
    - -
    -
    -property datafile: str | Path#
    -
    -
    Returns:
    -

    df – Absolute path to the file on disk where log_var (s) are stored during checkpointing or after solver has -stopped.

    -
    -
    Return type:
    -

    Path

    -
    -
    -
    - -
    -
    -busy()[source]#
    -

    Test if an async-running solver has stopped.

    -

    This method is only usable after calling fit() with mode=ASYNC. See -Solver for usage examples.

    -
    -
    Returns:
    -

    b – True if solver has stopped, False otherwise.

    -
    -
    Return type:
    -

    bool

    -
    -
    -
    - -
    -
    -solution()[source]#
    -

    Output the “solution” of the optimization problem.

    -

    This is a helper method intended for novice users. The return type is sub-class dependent, so don’t write an -API using this: use stats() instead.

    -
    - -
    -
    -stop()[source]#
    -

    Stop an async-running solver.

    -

    This method is only usable after calling fit() with mode=ASYNC. See -Solver for usage examples.

    -

    This method will block until the solver has stopped.

    -

    There is no guarantee that a checkpoint on disk exists once halted. (Reason: potential exceptions raised during -solver’s progress.) Users should invoke writeback() afterwards if needed.

    -

    Users must call this method to terminate an async-solver, even if busy() is False.

    -
    - -
    -
    -writeback()[source]#
    -

    Checkpoint state to disk.

    -
    - -
    -
    -default_stop_crit()[source]#
    -

    Default stopping criterion for solver if unspecified in fit() calls.

    -

    Sub-classes are expected to overwrite this method. If not overridden, then omitting the stop_crit parameter -in fit() is forbidden.

    -
    -
    Return type:
    -

    StoppingCriterion

    -
    -
    -
    - -
    -
    -objective_func()[source]#
    -

    Evaluate objective function given current math state.

    -

    The output array must have shape:

    -
      -
    • (1,) if evaluated at 1 point,

    • -
    • (N, 1) if evaluated at N different points.

    • -
    -

    Sub-classes are expected to overwrite this method. If not overridden, then enabling track_objective in -fit() is forbidden.

    -
    -
    Return type:
    -

    NDArray

    -
    -
    -
    - -
    - -
    -
    -class SolverMode(value, names=None, *, module=None, qualname=None, type=None, start=1, boundary=None)[source]#
    -

    Bases: Enum

    -

    Solver execution mode.

    -
    -
    -BLOCK = 1#
    -
    - -
    -
    -MANUAL = 2#
    -
    - -
    -
    -ASYNC = 3#
    -
    - -
    - -
    -
    -class StoppingCriterion[source]#
    -

    Bases: object

    -

    State machines (SM) which decide when to stop iterative solvers by examining their mathematical state.

    -

    SM decisions are always accompanied by at least one numerical statistic. These stats may be queried by solvers via -info() to provide diagnostic information to users.

    -

    Composite stopping criteria can be implemented via the overloaded (and[&], or[|]) operators.

    -
    -
    -stop(state)[source]#
    -

    Compute a stop signal based on the current mathematical state.

    -
    -
    Parameters:
    -

    state (Mapping) –

    Full mathematical state of solver at some iteration, i.e. _mstate.

    -

    Values from state may be cached inside the instance to form complex stopping conditions.

    -

    -
    -
    Returns:
    -

    s – True if no further iterations should be performed, False otherwise.

    -
    -
    Return type:
    -

    bool

    -
    -
    -
    - -
    -
    -info()[source]#
    -

    Get statistics associated with the last call to stop().

    -
    -
    Returns:
    -

    data

    -
    -
    Return type:
    -

    Mapping

    -
    -
    -
    - -
    -
    -clear()[source]#
    -

    Clear SM state (if any).

    -

    This method is useful when a StoppingCriterion instance must be reused in another call to -fit().

    -
    - -
    - -
    - - -
    - - - - - -
    - - - - - - -
    -
    - -
    - -
    -
    -
    - - - - - -
    - - -
    - - \ No newline at end of file diff --git a/api/experimental.sampler.html b/api/experimental.sampler.html index 79b2c7e1..a5d93123 100644 --- a/api/experimental.sampler.html +++ b/api/experimental.sampler.html @@ -18,8 +18,8 @@ - - + + @@ -32,7 +32,7 @@ - + @@ -736,13 +736,13 @@

    Remarks#< diff --git a/api/index.html b/api/index.html index ebe4ddcb..c9bdc0c1 100644 --- a/api/index.html +++ b/api/index.html @@ -28,8 +28,8 @@ - - + + @@ -44,7 +44,7 @@ - + @@ -833,7 +833,7 @@

    pyxu.math

    Stochastic trace estimation of a linear operator based on the Hutch++ algorithm.

    norm(x, **kwargs)

    -

    This function is identical to numpy.linalg.norm().

    +

    This function is identical to numpy.linalg.norm().

    trace(op[, xp, dtype])

    Exact trace of a linear operator based on multiple evaluation of the forward operator.

    @@ -858,7 +858,7 @@

    SciPy

    from_sciop(cls, sp_op)

    -

    Wrap a LinearOperator as a LinOp (or sub-class thereof).

    +

    Wrap a LinearOperator as a LinOp (or sub-class thereof).

    @@ -1620,7 +1620,7 @@

    Misc @@ -1628,7 +1628,7 @@

    Misc

    - Created with Sphinx 7.2.5, with some help from ChatGPT (GPT 4). + Created with Sphinx 7.2.6, with some help from ChatGPT (GPT 4).

    diff --git a/api/info.deps.html b/api/info.deps.html index e789b5d8..02e8958a 100644 --- a/api/info.deps.html +++ b/api/info.deps.html @@ -28,8 +28,8 @@ - - + + @@ -44,7 +44,7 @@ - + @@ -563,7 +563,7 @@

    Python module associated to an array backend.

    Parameters:
    -

    linalg (bool) – Return the linear-algebra submodule with identical API to numpy.linalg.

    +

    linalg (bool) – Return the linear-algebra submodule with identical API to numpy.linalg.

    Return type:

    module

    @@ -632,7 +632,7 @@

    Python module associated to an array backend.

    Parameters:
    -

    linalg (bool) – Return the linear-algebra submodule with identical API to scipy.sparse.linalg.

    +

    linalg (bool) – Return the linear-algebra submodule with identical API to scipy.sparse.linalg.

    Return type:

    module

    @@ -814,7 +814,7 @@ diff --git a/api/info.warning.html b/api/info.warning.html index 917a9fcc..d5e0008c 100644 --- a/api/info.warning.html +++ b/api/info.warning.html @@ -28,8 +28,8 @@ - - + + @@ -44,7 +44,7 @@ - + @@ -650,7 +650,7 @@ diff --git a/api/operator/blocks.html b/api/operator/blocks.html index 32f55618..5a1bdb18 100644 --- a/api/operator/blocks.html +++ b/api/operator/blocks.html @@ -28,8 +28,8 @@ - - + + @@ -44,7 +44,7 @@ - + @@ -871,7 +871,7 @@

    pyxu.operator.blocks @@ -879,7 +879,7 @@

    pyxu.operator.blocks

    - Created with Sphinx 7.2.5, with some help from ChatGPT (GPT 4). + Created with Sphinx 7.2.6, with some help from ChatGPT (GPT 4).

    diff --git a/api/operator/func.html b/api/operator/func.html index d8abfba0..1e1c977e 100644 --- a/api/operator/func.html +++ b/api/operator/func.html @@ -28,8 +28,8 @@ - - + + @@ -44,7 +44,7 @@ - + @@ -1218,7 +1218,7 @@

    Indicator Functions @@ -1226,7 +1226,7 @@

    Indicator Functions

    - Created with Sphinx 7.2.5, with some help from ChatGPT (GPT 4). + Created with Sphinx 7.2.6, with some help from ChatGPT (GPT 4).

    diff --git a/api/operator/index.html b/api/operator/index.html index f87fb003..32de64a6 100644 --- a/api/operator/index.html +++ b/api/operator/index.html @@ -28,8 +28,8 @@ - - + + @@ -44,7 +44,7 @@ - + @@ -579,7 +579,7 @@

    pyxu.operator @@ -587,7 +587,7 @@

    pyxu.operator

    - Created with Sphinx 7.2.5, with some help from ChatGPT (GPT 4). + Created with Sphinx 7.2.6, with some help from ChatGPT (GPT 4).

    diff --git a/api/operator/linop.html b/api/operator/linop.html index 537cf78e..bee48083 100644 --- a/api/operator/linop.html +++ b/api/operator/linop.html @@ -18,8 +18,8 @@ - - + + @@ -32,7 +32,7 @@ - + @@ -699,7 +699,7 @@

    Basic Operators<

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a/_images/opt-solver-2.png b/_images/opt-solver-2.png index 4a24cdaec2bca366e40296b73441a96334370519..d809bafa0be98e0f2bf6d9e2cc28b65e3a42426b 100644 GIT binary patch delta 45 zcmaFS&iJC8ae|wIg`R2PG*u`eo?x<>BgNc?0|0@T B5rY5# delta 45 zcmaFS&iJC8ae|wIxt@`ZLPkkRL9vy-er{q(K~8>2PG*u`eo?x%1OS delta 45 zcmex(l=0(H#tCi;=6Xgt3K=CO1;tkS`nicE1v&X8Ihjd%`9Pzb0RWdK B5#9g* diff --git a/_images/util-1.png b/_images/util-1.png index 75581c81e99065a722a595c96b23686e15706b8d..f538455150752d7ca206506a3e77f400eb23af44 100644 GIT binary patch delta 45 zcmcb;j_LL~rU`Be7J3Fc3K=CO1;tkS`nicE1v&X8Ihjd%`9y delta 45 zcmcb;j_LL~rU`Be=6Xgt3K=CO1;tkS`nicE1v&X8Ihjd%`9Q1^1^|g< B5+ncs diff --git a/_images/util-2.png b/_images/util-2.png index e1df26600b3b8912832ed0fa356aa9c18b99ebb4..268536af7dc66377fb2be22ca7e7ab756a2cb9f3 100644 GIT binary patch delta 48 zcmdn~mSy`}mI-bO7J3Fc3K=CO1;tkS`nicE1v&X8Ihjd%`9 - - + + @@ -43,7 +43,7 @@ - + @@ -505,7 +505,7 @@

    Source code for collections.abc

             

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  • gpu (bool) – If True the singular value decomposition is performed on the GPU.

  • -
  • kwargs (Mapping) – Additional kwargs accepted by svds().

  • +
  • kwargs (Mapping) – Additional kwargs accepted by svds().

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  • tuple[str, …]: pad dimension[k] using mode[k].

  • -

    (See numpy.pad() for details.)

    +

    (See numpy.pad() for details.)

    @@ -860,8 +860,8 @@

    TransformsIf False, then apply() takes (…, 2N.prod()) inputs, i.e. \(\mathbb{C}^{N}\) vectors viewed as bijections with \(\mathbb{R}^{2N}\).

    -
  • kwargs (dict) –

    Extra kwargs passed to scipy.fft.fftn() or cupyx.scipy.fft.fftn().

    -

    Supported parameters for scipy.fft.fftn() are:

    +
  • kwargs (dict) –

    Extra kwargs passed to scipy.fft.fftn() or cupyx.scipy.fft.fftn().

    +

    Supported parameters for scipy.fft.fftn() are:

    -

    (See numpy.pad() for details.)

    +

    (See numpy.pad() for details.)

  • enable_warnings (bool) – If True, emit a warning in case of precision mis-match issues.

  • @@ -2615,7 +2620,7 @@

    Filters
  • tuple[str, …]: the d-th dimension uses mode[d] as boundary condition.

  • -

    (See numpy.pad() for details.)

    +

    (See numpy.pad() for details.)

  • gpu (bool) – Input NDArray type (True for GPU, False for CPU). Defaults to False.

  • dtype (DType) – Working precision of the linear operator.

  • @@ -2699,7 +2704,7 @@

    Filters
  • tuple[str, …]: the d-th dimension uses mode[d] as boundary condition.

  • -

    (See numpy.pad() for details.)

    +

    (See numpy.pad() for details.)

  • sampling (Real, list[Real]) – Sampling step (i.e. distance between two consecutive elements of an array). Defaults to 1.

  • @@ -2797,7 +2802,7 @@

    Filters
  • tuple[str, …]: the d-th dimension uses mode[d] as boundary condition.

  • -

    (See numpy.pad() for details.)

    +

    (See numpy.pad() for details.)

  • sampling (Real, list[Real]) – Sampling step (i.e. distance between two consecutive elements of an array). Defaults to 1.

  • @@ -2851,7 +2856,7 @@

    Filters

    mode (str | Sequence[str]) –

  • sampling (Real | Sequence[Real, ...]) –

  • gpu (bool) –

  • -
  • dtype (dtype[Any] | None | Type[Any] | _SupportsDType[dtype[Any]] | str | Tuple[Any, int] | Tuple[Any, SupportsIndex | Sequence[SupportsIndex]] | List[Any] | _DTypeDict | Tuple[Any, Any]) –

  • +
  • dtype (dtype[Any] | None | type[Any] | _SupportsDType[dtype[Any]] | str | tuple[Any, int] | tuple[Any, SupportsIndex | Sequence[SupportsIndex]] | list[Any] | _DTypeDict | tuple[Any, Any]) –

  • @@ -2885,7 +2890,7 @@

    Filters
  • tuple[str, …]: the d-th dimension uses mode[d] as boundary condition.

  • -

    (See numpy.pad() for details.)

    +

    (See numpy.pad() for details.)

  • sampling (Real, list[Real]) – Sampling step (i.e. distance between two consecutive elements of an array). Defaults to 1.

  • @@ -2961,7 +2966,7 @@

    Filters
  • tuple[str, …]: the d-th dimension uses mode[d] as boundary condition.

  • -

    (See numpy.pad() for details.)

    +

    (See numpy.pad() for details.)

  • sampling (Real, list[Real]) – Sampling step (i.e. distance between two consecutive elements of an array). Defaults to 1.

  • @@ -3037,7 +3042,7 @@

    Filters
  • tuple[str, …]: the d-th dimension uses mode[d] as boundary condition.

  • -

    (See numpy.pad() for details.)

    +

    (See numpy.pad() for details.)

  • sampling (Real, list[Real]) – Sampling step (i.e. distance between two consecutive elements of an array). Defaults to 1.

  • @@ -3113,7 +3118,7 @@

    Filters
  • tuple[str, …]: the d-th dimension uses mode[d] as boundary condition.

  • -

    (See numpy.pad() for details.)

    +

    (See numpy.pad() for details.)

  • sampling (Real, list[Real]) – Sampling step (i.e. distance between two consecutive elements of an array). Defaults to 1.

  • @@ -3313,7 +3318,7 @@

    Remark#

    mode (str | Sequence[str]) –

  • sampling (Real | tuple[Real, ...]) –

  • gpu (bool) –

  • -
  • dtype (dtype[Any] | None | Type[Any] | _SupportsDType[dtype[Any]] | str | Tuple[Any, int] | Tuple[Any, SupportsIndex | Sequence[SupportsIndex]] | List[Any] | _DTypeDict | Tuple[Any, Any]) –

  • +
  • dtype (dtype[Any] | None | type[Any] | _SupportsDType[dtype[Any]] | str | tuple[Any, int] | tuple[Any, SupportsIndex | Sequence[SupportsIndex]] | list[Any] | _DTypeDict | tuple[Any, Any]) –

  • parallel (bool) –

  • @@ -3402,7 +3407,7 @@

    Derivatives
  • tuple[str, …]: the d-th dimension uses mode[d] as boundary condition.

  • -

    (See numpy.pad() for details.)

    +

    (See numpy.pad() for details.)

  • gpu (bool) – Input NDArray type (True for GPU, False for CPU). Defaults to False.

  • @@ -3584,7 +3589,7 @@

    Derivatives
  • tuple[str, …]: the d-th dimension uses mode[d] as boundary condition.

  • -

    (See numpy.pad() for details.)

    +

    (See numpy.pad() for details.)

  • gpu (bool) – Input NDArray type (True for GPU, False for CPU). Defaults to False.

  • @@ -3735,7 +3740,7 @@

    Derivatives
  • tuple[str, …]: the d-th dimension uses mode[d] as boundary condition.

  • -

    (See numpy.pad() for details.)

    +

    (See numpy.pad() for details.)

  • gpu (bool) – Input NDArray type (True for GPU, False for CPU). Defaults to False.

  • @@ -3856,7 +3861,7 @@

    Derivatives
  • tuple[str, …]: the d-th dimension uses mode[d] as boundary condition.

  • -

    (See numpy.pad() for details.)

    +

    (See numpy.pad() for details.)

  • gpu (bool) – Input NDArray type (True for GPU, False for CPU). Defaults to False.

  • @@ -3959,7 +3964,7 @@

    Derivatives
  • tuple[str, …]: the d-th dimension uses mode[d] as boundary condition.

  • -

    (See numpy.pad() for details.)

    +

    (See numpy.pad() for details.)

  • gpu (bool) – Input NDArray type (True for GPU, False for CPU). Defaults to False.

  • @@ -4088,7 +4093,7 @@

    Derivatives
  • tuple[str, …]: the d-th dimension uses mode[d] as boundary condition.

  • -

    (See numpy.pad() for details.)

    +

    (See numpy.pad() for details.)

  • gpu (bool) – Input NDArray type (True for GPU, False for CPU). Defaults to False.

  • @@ -4201,7 +4206,7 @@

    Derivatives
  • tuple[str, …]: the d-th dimension uses mode[d] as boundary condition.

  • -

    (See numpy.pad() for details.)

    +

    (See numpy.pad() for details.)

  • gpu (bool) – Input NDArray type (True for GPU, False for CPU). Defaults to False.

  • @@ -4347,7 +4352,7 @@

    Derivatives
  • tuple[str, …]: the d-th dimension uses mode[d] as boundary condition.

  • -

    (See numpy.pad() for details.)

    +

    (See numpy.pad() for details.)

  • parallel (bool) – If True, use Dask to evaluate the different partial derivatives in parallel.

  • diff_kwargs (dict) – Keyword arguments to parametrize partial derivatives (see @@ -4452,7 +4457,7 @@

    Derivatives
  • tuple[str, …]: the d-th dimension uses mode[d] as boundary condition.

  • -

    (See numpy.pad() for details.)

    +

    (See numpy.pad() for details.)

  • parallel (bool) – If True, use Dask to evaluate the different partial derivatives in parallel.

  • diff_kwargs (dict) – Keyword arguments to parametrize partial derivatives (see @@ -4572,7 +4577,7 @@

    Derivatives
  • tuple[str, …]: the d-th dimension uses mode[d] as boundary condition.

  • -

    (See numpy.pad() for details.)

    +

    (See numpy.pad() for details.)

  • parallel (bool) – If True, use Dask to evaluate the different partial derivatives in parallel.

  • diff_kwargs (dict) – Keyword arguments to parametrize partial derivatives (see @@ -4690,7 +4695,7 @@

    Derivatives
  • tuple[str, …]: the d-th dimension uses mode[d] as boundary condition.

  • -

    (See numpy.pad() for details.)

    +

    (See numpy.pad() for details.)

  • parallel (bool) – If True, use Dask to evaluate the different partial derivatives in parallel.

  • diff_kwargs (dict) – Keyword arguments to parametrize partial derivatives (see @@ -5043,13 +5048,13 @@

    Tensor Products< diff --git a/api/operator/map.html b/api/operator/map.html index 106ffe8d..db3ae0a8 100644 --- a/api/operator/map.html +++ b/api/operator/map.html @@ -28,8 +28,8 @@ - - + + @@ -44,7 +44,7 @@ - + @@ -1338,7 +1338,7 @@

    Misc @@ -1346,7 +1346,7 @@

    Misc

    - Created with Sphinx 7.2.5, with some help from ChatGPT (GPT 4). + Created with Sphinx 7.2.6, with some help from ChatGPT (GPT 4).

    diff --git a/api/opt.solver.html b/api/opt.solver.html index cac93364..906e4d8a 100644 --- a/api/opt.solver.html +++ b/api/opt.solver.html @@ -18,8 +18,8 @@ - - + + @@ -32,7 +32,7 @@ - + @@ -1934,13 +1934,13 @@

    Parameters (fit diff --git a/api/opt.stop.html b/api/opt.stop.html index a535ac8b..4f9d299e 100644 --- a/api/opt.stop.html +++ b/api/opt.stop.html @@ -28,8 +28,8 @@ - - + + @@ -44,7 +44,7 @@ - + @@ -821,7 +821,7 @@

    pyxu.opt.stop @@ -829,7 +829,7 @@

    pyxu.opt.stop

    - Created with Sphinx 7.2.5, with some help from ChatGPT (GPT 4). + Created with Sphinx 7.2.6, with some help from ChatGPT (GPT 4).

    diff --git a/api/runtime.html b/api/runtime.html index 6e6f72fb..5b8e9c96 100644 --- a/api/runtime.html +++ b/api/runtime.html @@ -28,8 +28,8 @@ - - + + @@ -44,7 +44,7 @@ - + @@ -787,7 +787,7 @@ diff --git a/api/util/array_module.html b/api/util/array_module.html deleted file mode 100644 index 978e80f2..00000000 --- a/api/util/array_module.html +++ /dev/null @@ -1,746 +0,0 @@ - - - - - - - - - - - pyxu.util.array_module — Pyxu Documentation - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
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    - - - -
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    - -
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    pyxu.util.array_module#

    -
    -
    -get_array_module(x, fallback=None)[source]#
    -

    Get the array namespace corresponding to a given object.

    -
    -
    Parameters:
    -
      -
    • x (object) – Any object compatible with the interface of NumPy arrays.

    • -
    • fallback (ArrayModule) – Fallback module if x is not a NumPy-like array. Default behaviour: raise error if fallback used.

    • -
    -
    -
    Returns:
    -

    namespace – The namespace to use to manipulate x, or fallback if provided.

    -
    -
    Return type:
    -

    ArrayModule

    -
    -
    -
    - -
    -
    -redirect(i, **kwargs)[source]#
    -

    Change codepath for supplied array backends.

    -

    Some functions/methods cannot be written in module-agnostic fashion. The action of this decorator is summarized -below:

    -
      -
    • Analyze an array-valued parameter (x) of the wrapped function/method (f).

    • -
    • If x lies in one of the supplied array namespaces: re-route execution to the specified function.

    • -
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    • -
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    Parameters:
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    Callable

    -
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    -

    Notes

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    Auto-dispatch via redirect() assumes the dispatcher/dispatchee have the same parameterization, i.e.:

    -
      -
    • if f is a function -> dispatch possible to another callable with identical signature (i.e., function or -staticmethod)

    • -
    • if f is a staticmethod -> dispatch possible to another callable with identical signature (i.e., function or -staticmethod)

    • -
    • if f is an instance-method -> dispatch to another instance-method of the class with identical signature.

    • -
    -

    Example

    -
    def f(x, y): return "f"
    -
    -@redirect('x', NUMPY=f)    # if 'x' is of type NDArrayInfo.NUMPY, i.e. has
    -def g(x, y): return "g"    # short-name 'NUMPY' -> reroute execution to `f`
    -
    -x1 = np.arange(5)
    -x2 = da.array(x1)
    -y = 1
    -g(x1, y), g(x2, y)  # 'f', 'g'
    -
    -
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    - -
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    -compute(*args, mode='compute', **kwargs)[source]#
    -

    Force computation of Dask collections.

    -
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    Parameters:
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      -
    • *args (object, list) – Any number of objects. If it is a dask object, it is evaluated and the result is returned. Non-dask arguments -are passed through unchanged. Python collections are traversed to find/evaluate dask objects within. (Use -traverse =False to disable this behavior.)

    • -
    • mode (str) – Dask evaluation strategy: compute or persist.

    • -
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    -to_NUMPY(x)[source]#
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    Convert an array from a specific backend to NUMPY.

    -
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    Parameters:
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    x (NDArray) – Array to be converted.

    -
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    Returns:
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    pyxu.util.complex#

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    -
    -view_as_complex_mat(rmat, real_input=False, real_output=False)[source]#
    -

    View real-valued matrix as its complex-valued equivalent. (Inverse of view_as_real_mat().)

    -

    Useful to transform real-valued matrix/vector products to their complex-valued counterparts.

    -
    -
    Parameters:
    -
      -
    • rmat (NDArray) –

      Real-valued matrix. Accepted dimensions depend on the values of real_input and real_output:

      -
      | real_input | real_output | rmat.shape |
      -|------------|-------------|------------|
      -| False      | False       | (2M, 2N)   |
      -| False      | True        | ( M, 2N)   |
      -| True       | False       | (2M,  N)   |
      -| True       | True        | ( M,  N)   |
      -
      -
      -

    • -
    • real_input (bool) –

    • -
    • real_output (bool) –

    • -
    -
    -
    Returns:
    -

    cmat – (M, N) complex-valued matrix.

    -
    -
    Return type:
    -

    NDArray

    -
    -
    -

    Examples

    -
    from pyxu.util import view_as_real_mat, view_as_complex_mat
    -
    -A = np.array([[ 0., -2.,  1., -3.,  2., -4.],
    -              [ 2.,  0.,  3.,  1.,  4.,  2.],
    -              [ 3., -5.,  4., -6.,  5., -7.],
    -              [ 5.,  3.,  6.,  4.,  7.,  5.]])
    -
    -B = view_as_complex(A)  # array([[0.+2.j, 1.+3.j, 2.+4.j],
    -                        #        [3.+5.j, 4.+6.j, 5.+7.j]])
    -
    -
    -

    Notes

    -
      -
    • Complex-valued matrices are returned unchanged.

    • -
    • Real-valued matrices are mapped into a complex-valued matrix \({A}\in\mathbb{C}^{M\times N}\) as follows:

      -
        -
      • \(\hat{A}\in\mathbb{R}^{2M\times 2N}\): \(A_{m,n} = \hat{A}_{2m-1,2n-1} + j \hat{A}_{2m,2n-1}\),

      • -
      • \(\hat{A}\in\mathbb{R}^{M\times 2N}\): \(A_{m,n} = \hat{A}_{m,2n-1} - j \hat{A}_{m,2n}\),

      • -
      • \(\hat{A}\in\mathbb{R}^{2M\times N}\): \(A_{m,n} = \hat{A}_{2m-1,n} + j \hat{A}_{2m,n}\),

      • -
      • \(\hat{A}\in\mathbb{R}^{M\times N}\): \(A_{m,n} = \hat{A}_{m,n} + 0j\),

      • -
      -

      for \(1\leq n \leq N\), \(1\leq m\leq M\).

      -
    • -
    - -
    - -
    -
    -view_as_complex(x)[source]#
    -

    View real-valued array as its complex-valued bijection. (Inverse of view_as_real().)

    -
    -
    Parameters:
    -

    x (NDArray) – (…, 2N) real-valued array.

    -
    -
    Returns:
    -

    y – (…, N) complex-valued array.

    -
    -
    Return type:
    -

    NDArray

    -
    -
    -

    Examples

    -
    from pyxu.util import view_as_real, view_as_complex
    -x = np.arange(6.0)      # array([0., 1., 2., 3., 4., 5.])
    -y = view_as_complex(x)  # array([0.+1.j, 2.+3.j, 4.+5.j])
    -view_as_real(y) == x    # True
    -
    -
    -

    Notes

    -

    Complex-valued inputs are returned unchanged. For real-valued inputs, this function acts on the last axis as:

    -
    -\[y_n = x_{2n-1}+j \, x_{2n}, \qquad 1\leq n\leq N.\]
    - -
    - -
    -
    -view_as_real_mat(cmat, real_input=False, real_output=False)[source]#
    -

    View complex-valued matrix as its real-valued equivalent. (Inverse of view_as_complex_mat().)

    -

    Useful to transform complex-valued matrix/vector products to their real-valued counterparts.

    -
    -
    Parameters:
    -
      -
    • cmat (NDArray) – (M, N) complex-valued matrix.

    • -
    • real_input (bool) –

    • -
    • real_output (bool) –

    • -
    -
    -
    Returns:
    -

    rmat

    -

    The output shape depends on the values of real_input and real_output:

    -
    | real_input | real_output | rmat.shape |
    -|------------|-------------|------------|
    -| False      | False       | (2M, 2N)   |
    -| False      | True        | ( M, 2N)   |
    -| True       | False       | (2M,  N)   |
    -| True       | True        | ( M,  N)   |
    -
    -
    -

    -
    -
    Return type:
    -

    NDArray

    -
    -
    -

    Examples

    -
    from pyxu.util import view_as_real_mat, view_as_complex_mat
    -
    -A = np.reshape(
    -    np.r_[:6] + 1j * np.r_[2:8], # array([[0.+2.j, 1.+3.j, 2.+4.j],
    -    newshape=(2, 3),             #        [3.+5.j, 4.+6.j, 5.+7.j]])
    -)
    -B = view_as_real_mat(A)          # array([[ 0., -2.,  1., -3.,  2., -4.],
    -                                 #        [ 2.,  0.,  3.,  1.,  4.,  2.],
    -                                 #        [ 3., -5.,  4., -6.,  5., -7.],
    -                                 #        [ 5.,  3.,  6.,  4.,  7.,  5.]])
    -
    -
    -

    Notes

    -
      -
    • Real-valued matrices are returned unchanged.

    • -
    • Complex-valued matrices \(A\in\mathbb{C}^{M\times N}\) are mapped into a real-valued matrix -\(\hat{A}\in\mathbb{R}^{2M\times 2N}\) defined, for \(1\leq n \leq N\), \(1\leq m\leq M\) as

      -
      -\[\begin{split}\hat{A}_{2m-1,2n-1} = \mathcal{R}(A_{m,n}), -& \quad -\hat{A}_{2m-1,2n} = -\mathcal{I}(A_{m,n}),\\ -\hat{A}_{2m,2n-1} = \mathcal{I}(A_{m,n}), -& \quad -\hat{A}_{2m,2n}=\mathcal{R}(A_{m,n}).\end{split}\]
      -

      If real_[in|out]put=True, then even columns/rows (or both) are furthermore dropped. We have view_as_real(A -@ x) = view_as_real_mat(A) @ view_as_real(x).

      -
    • -
    - -
    - -
    -
    -view_as_real(x)[source]#
    -

    View complex-valued array as its real-valued bijection. (Inverse of view_as_complex().)

    -
    -
    Parameters:
    -

    x (NDArray) – (…, N) complex-valued array.

    -
    -
    Returns:
    -

    y – (…, 2N) real-valued array.

    -
    -
    Return type:
    -

    NDArray

    -
    -
    -

    Examples

    -
    from pyxu.util import view_as_real, view_as_complex
    -x = np.r_[:3] + 1j * np.r_[2:5]   # array([0.+2.j, 1.+3.j, 2.+4.j])
    -y = view_as_real(x)               # array([0., 2., 1., 3., 2., 4.])
    -view_as_complex(y) == x           # True
    -
    -
    -

    Notes

    -

    Real-valued inputs are returned unchanged. For complex-valued inputs, this function acts on the last axis as:

    -
    -\[y_{2n-1} = \mathcal{R}(x_n), -\quad -y_{2n} = \mathcal{I}(x_n), -\quad 1\leq n\leq N,\]
    -

    where \(\mathcal{R}, \mathcal{I}\) denote the real/imaginary parts respectively.

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    -copy_if_unsafe(x)[source]#
    -

    Copy array if it is unsafe to do in-place updates on it.

    -

    In-place updates are unsafe if:

    -
      -
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    • -
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    • -
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    -import_module(name, fail_on_error=True)[source]#
    -

    Load a Python module dynamically.

    -
    -
    Parameters:
    -
      -
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    • -
    • fail_on_error (bool) –

    • -
    -
    -
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    -

    module

    -
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    -
    -infer_composition_shape(sh1, sh2)[source]#
    -

    Infer shape of arithmetic operation \(A \circ B\).

    -
    -
    Parameters:
    -
    -
    -
    Return type:
    -

    tuple[Integral, Integral]

    -
    -
    -
    - -
    -
    -infer_sum_shape(sh1, sh2)[source]#
    -

    Infer shape of arithmetic operation \(A + B\).

    -
    -
    Parameters:
    -
    -
    -
    Return type:
    -

    tuple[Integral, Integral]

    -
    -
    -
    - -
    -
    -next_fast_len(N, even=False)[source]#
    -

    Compute the next 5-smooth number greater-or-equal to N.

    -

    If even=True, then ensure the next 5-smooth number is even-valued.

    -
    -
    Parameters:
    -
    -
    -
    Return type:
    -

    Integral

    -
    -
    -
    - -
    -
    -parse_params(func, *args, **kwargs)[source]#
    -

    Get function parameterization.

    -
    -
    Returns:
    -

    params – (key, value) params as seen in body of func when called via func(*args, **kwargs).

    -
    -
    Return type:
    -

    Mapping

    -
    -
    -
    - -
    -
    -read_only(x)[source]#
    -

    Make an array read-only.

    -
    -
    Parameters:
    -

    x (NDArray) –

    -
    -
    Returns:
    -

    y

    -
    -
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    -

    NDArray

    -
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    -peaks(x, y)[source]#
    -

    Matlab 2D peaks function.

    -

    Peaks is a function of two variables, obtained by translating and scaling Gaussian distributions. (See Matlab’s -peaks function.)

    -

    This function is useful for testing purposes.

    -
    -
    Parameters:
    -
    -
    -
    Returns:
    -

    z – Values of the 2D function peaks at the points specified by the entries of x and y.

    -
    -
    Return type:
    -

    NDArray

    -
    -
    -

    Examples

    -
    import numpy as np
    -import matplotlib.pyplot as plt
    -from pyxu.util.misc import peaks
    -
    -x = np.linspace(-3, 3, 1000)
    -xx, yy = np.meshgrid(x, x)
    -z = peaks(xx, yy)
    -plt.figure()
    -plt.imshow(z)
    -
    -
    -

    (Source code, png, hires.png, pdf)

    -
    -../../_images/misc-1.png -
    -
    - -
    -
    -star_like_sample(N, w, s, po, x0, ndi=NDArrayInfo.NUMPY)[source]#
    -

    Star-like test image.

    -

    Generates a (N, N) square image of a star-like object normalized between 0 and 1. Based on GlobalBioIm’s -StarLikeSample function. This function is -useful for testing purposes as it contains high-frequency information.

    -
    -
    Parameters:
    -
      -
    • N (Integer) – Size of the image (must be an even number).

    • -
    • w (Integer) – The number of branches of the sample will be 4*w.

    • -
    • s (Real) – Slope of the sigmoid function \(\frac1{1+\exp[s (x-x_{0})]}\) attenuating the boundaries.

    • -
    • po (Integer) – Power-raising factor for the final image (to have smoother edges).

    • -
    • x0 (Real) – Radial shift of the sigmoid function \(\frac1{1+\exp[s (x-x_{0})]}\).

    • -
    • ndi (NDArrayInfo) – Desired array module for the output.

    • -
    -
    -
    Returns:
    -

    image – (N, N) image of star-like sample.

    -
    -
    Return type:
    -

    NDArray

    -
    -
    -

    Examples

    -
    import numpy as np
    -import matplotlib.pyplot as plt
    -from pyxu.util.misc import star_like_sample
    -
    -star = star_like_sample(N=256, w=8, s=20, po=3, x0=0.7)
    -plt.figure()
    -plt.imshow(star)
    -
    -
    -

    (Source code, png, hires.png, pdf)

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    -as_canonical_shape(x)[source]#
    -

    Transform a lone integer into a valid tuple-based shape specifier.

    -
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    -

    x (Integral | tuple[Integral, ...]) –

    -
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    -

    Integral | tuple[Integral, …]

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    -vectorize(i, method='scan', codim=None)[source]#
    -

    Decorator to auto-vectorize an array function to abide by Property API rules.

    -
    -
    Parameters:
    -
      -
    • i (VarName) – Function parameter to vectorize. This variable must hold an object with a NumPy API.

    • -
    • method (str) –

      Vectorization strategy:

      -
        -
      • scan computes outputs using a for-loop.

      • -
      • parallel passes inputs to DASK and evaluates them in parallel.

      • -
      • scan_dask passes inputs to DASK but evaluates inputs in sequence. -This is useful if the function being vectorized has a shared resource, i.e. is not thread-safe. It -effectively gives a DASK-unaware function the ability to work with DASK inputs.

      • -
      -

    • -
    • codim (Integer) –

      Size of the function’s core dimension output.

      -

      This parameter is only required in “parallel” and “scan_dask” modes.

      -

    • -
    -
    -
    Return type:
    -

    Callable

    -
    -
    -

    Example

    -
    import pyxu.util as pxu
    -
    -@pxu.vectorize('x')
    -def f(x):
    -    return x.sum(keepdims=True)
    -
    -x = np.arange(10).reshape((2, 5))
    -f(x[0]), f(x[1])  #  [10], [35]
    -f(x)              #  [10, 35] -> would have retured [45] if not decorated.
    -
    -
    -
    - -
    -
    -_array_ize(data, shape, dtype)[source]#
    -

    Transform a Dask-delayed object into its Dask-array counterpart.

    -

    This function is a no-op if data is not a Delayed object.

    -
    -
    Parameters:
    -
      -
    • data (NDArray, Delayed) – Object to act upon.

    • -
    • shape (NDArrayShape) –

      Shape of data.

      -

      This parameter is only used if data is a Delayed object. Its goal is to transform -the former back to array form.

      -

    • -
    • dtype (DType) –

      Dtype of data.

      -

      This parameter is only used if data is a Delayed object. Its goal is to transform -the former back to array form.

      -

    • -
    -
    -
    Returns:
    -

    arr – Dask-backed NDArray if data was a Delayed object; no-op otherwise.

    -
    -
    Return type:
    -

    NDArray

    -
    -
    -
    - -
    -
    -_dask_zip(func, data, out_shape, out_dtype, parallel)[source]#
    -

    Map functions in parallel via Dask.

    -

    Computes the equivalent of out = [f(x) for (f, x) in zip(func, data)], with the following semantics:

    -
      -
    • If data contains only NUMPY/CUPY arrays, then out is computed as above.

    • -
    • If data contains only DASK arrays, then entries of out are computed:

      -
        -
      • if parallel enabled -> dask-delay each func, then evaluate in parallel.

      • -
      • if parallel disabled -> dask-delay each func, then evaluate in sequence. -(This is useful if func s share a common resource, thus not thread-safe.)

      • -
      -
    • -
    -

    For Dask-array inputs, this amounts to creating a task graph with virtual dependencies between successive func -calls. In other words, the task graph looks like:

    -
    _dask_zip(func, data, parallel=True) -> out
    -
    -              +----+              +----+
    -    data[0]-->|func|-->blk[0]-+-->|list|-->out
    -        .     +----+          |   +----+
    -        .                     |
    -        .     +----+          |
    -    data[n]-->|func|-->blk[n]-+
    -              +----+
    -
    -==========================================================================================================
    -_dask_zip(func, data, parallel=False) -> out
    -                                                                                       +----+
    -                        +-------------------+----------------+--------------------+----+list|-->out
    -                        |                   |                |                    |    +----+
    -                        |                   |                |                    |
    -              +----+    |        +----+     |      +---+     |         +----+     |
    -    data[0]-->|func|-->out[0]-+->|func|-->out[1]-->|...|-->out[n-1]-+->|func|-->out[n]
    -              +----+          |  +----+            +---+            |  +----+
    -                              |                                     |
    -    data[1]-------------------+                                     |
    -        .                                                           |
    -        .                                                           |
    -        .                                                           |
    -    data[n]---------------------------------------------------------+
    -
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    Parameters:
    -
      -
    • func (list) –

      Functions to apply to each element of data.

      -

      Function signatures are Callable[[NDArray], NDArray].

      -

    • -
    • data (list) – (N_data,) arrays to act on.

    • -
    • out_shape (list) –

      Shapes of func[i](data[i]).

      -

      This parameter is only used if inputs are DASK arrays. Its goal is to transform -Delayed objects back to array form.

      -

    • -
    • out_dtype (list) –

      Dtypes of func[i](data[i]).

      -

      This parameter is only used if inputs are DASK arrays. Its goal is to transform -Delayed objects back to array form.

      -

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    • parallel (bool) –

    • -
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    -

    out – (N_data,) objects acted upon.

    -

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    -

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    - - -
    - - \ No newline at end of file diff --git a/examples/contributing.html b/examples/contributing.html index 11edaac8..bd3909b9 100644 --- a/examples/contributing.html +++ b/examples/contributing.html @@ -18,8 +18,8 @@ - - + + @@ -32,7 +32,7 @@ - + @@ -416,13 +416,13 @@

    Conclusion diff --git a/examples/index.html b/examples/index.html index 68c9c467..b063cc58 100644 --- a/examples/index.html +++ b/examples/index.html @@ -28,8 +28,8 @@ - - + + @@ -38,14 +38,14 @@ - + - + @@ -568,7 +568,7 @@

    Example Gallery @@ -576,7 +576,7 @@

    Example Gallery

    - Created with Sphinx 7.2.5, with some help from ChatGPT (GPT 4). + Created with Sphinx 7.2.6, with some help from ChatGPT (GPT 4).

    diff --git a/examples/stencils.html b/examples/stencils.html index b4eed9bb..b7375665 100644 --- a/examples/stencils.html +++ b/examples/stencils.html @@ -18,8 +18,8 @@ - - + + @@ -28,12 +28,12 @@ - + - + @@ -488,19 +488,19 @@

    Scico#< import jax.numpy as jnp # Convert data and kernel from Numpy to Jax -data_jax = jnp.asarray(data) -k2d_jax = jnp.asarray(kernel_2d) +data_jax = jnp.asarray(data) +k2d_jax = jnp.asarray(kernel_2d) config.update("jax_enable_x64", True) # 2d convolution conv_scico = Conv_scico( - k2d_jax, + k2d_jax, input_shape=data_jax.shape, input_dtype=data_jax.dtype, mode="same", jit=True, ) -y_scico = conv_scico(data_jax) +y_scico = conv_scico(data_jax) assert np.allclose(y, y_scico) @@ -510,23 +510,23 @@

    Scico#<
    # Separable convolution
    -k1d_jax = jnp.asarray(kernel_1d)
    +k1d_jax = jnp.asarray(kernel_1d)
     
     conv_scico_rows = Conv_scico(
    -    k1d_jax[np.newaxis, :],
    +    k1d_jax[np.newaxis, :],
         input_shape=data_jax.shape,
         input_dtype=data_jax.dtype,
         mode="same",
         jit=True,
     )
     conv_scico_cols = Conv_scico(
    -    k1d_jax[:, np.newaxis],
    +    k1d_jax[:, np.newaxis],
         input_shape=data_jax.shape,
         input_dtype=data_jax.dtype,
         mode="same",
         jit=True,
     )
    -y_scico = conv_scico_cols(conv_scico_rows(data_jax))
    +y_scico = conv_scico_cols(conv_scico_rows(data_jax))
     assert np.allclose(y, y_scico)
     
    @@ -567,8 +567,8 @@

    Separable 1d convolutionfor j, width in enumerate(nwidths): if npix > (width * 2): kernel = np.fromfunction(gauss, (width,)) - data_jax = jnp.asarray(data) - k1d_jax = jnp.asarray(kernel_1d) + data_jax = jnp.asarray(data) + k1d_jax = jnp.asarray(kernel_1d) ## PYXU @@ -674,27 +674,27 @@

    Separable 1d convolutionconfig.update("jax_enable_x64", True) conv_scico_rows = Conv_scico( - k1d_jax[np.newaxis, :], + k1d_jax[np.newaxis, :], input_shape=data_jax.shape, input_dtype=data_jax.dtype, mode="same", jit=True, ) conv_scico_cols = Conv_scico( - k1d_jax[:, np.newaxis], + k1d_jax[:, np.newaxis], input_shape=data_jax.shape, input_dtype=data_jax.dtype, mode="same", jit=True, ) - y_scico = conv_scico_cols(conv_scico_rows(data_jax)) + y_scico = conv_scico_cols(conv_scico_rows(data_jax)) t_scico[0, i, j] = time.perf_counter() - tic # Run time times = 0 for _ in range(nreps): tic = time.perf_counter() - y_scico = conv_scico_cols(conv_scico_rows(data_jax)) + y_scico = conv_scico_cols(conv_scico_rows(data_jax)) times += time.perf_counter() - tic t_scico[1, i, j] = times / nreps pbar.update(1) @@ -924,13 +924,13 @@

    Direct 2d convolution diff --git a/fair/contribute.html b/fair/contribute.html index e6b9c540..49e965bf 100644 --- a/fair/contribute.html +++ b/fair/contribute.html @@ -28,8 +28,8 @@ - - + + @@ -44,7 +44,7 @@ - + @@ -429,7 +429,6 @@
  • PycSphere
  • pycWavelet
  • PYFW
  • -
  • TokemakRec
  • TVDenoiser
  • UncertaintyQuant
  • WaveProp
  • @@ -831,7 +830,7 @@

    Create your documentation @@ -839,7 +838,7 @@

    Create your documentation

    - Created with Sphinx 7.2.5, with some help from ChatGPT (GPT 4). + Created with Sphinx 7.2.6, with some help from ChatGPT (GPT 4).

    diff --git a/fair/dev_notes.html b/fair/dev_notes.html index a259943d..ad1db8e2 100644 --- a/fair/dev_notes.html +++ b/fair/dev_notes.html @@ -28,8 +28,8 @@ - - + + @@ -44,7 +44,7 @@ - + @@ -429,7 +429,6 @@
  • PycSphere
  • pycWavelet
  • PYFW
  • -
  • TokemakRec
  • TVDenoiser
  • UncertaintyQuant
  • WaveProp
  • @@ -652,7 +651,7 @@

    Example of a rule-abiding operator @@ -660,7 +659,7 @@

    Example of a rule-abiding operator

    - Created with Sphinx 7.2.5, with some help from ChatGPT (GPT 4). + Created with Sphinx 7.2.6, with some help from ChatGPT (GPT 4).

    diff --git a/fair/howto.html b/fair/howto.html index b53812bf..7d03930e 100644 --- a/fair/howto.html +++ b/fair/howto.html @@ -18,8 +18,8 @@ - - + + @@ -32,7 +32,7 @@ - + @@ -246,7 +246,6 @@
  • PycSphere
  • pycWavelet
  • PYFW
  • -
  • TokemakRec
  • TVDenoiser
  • UncertaintyQuant
  • WaveProp
  • @@ -428,13 +427,13 @@

    For Developers diff --git a/fair/index.html b/fair/index.html index e19ee23b..51c0a3c2 100644 --- a/fair/index.html +++ b/fair/index.html @@ -28,8 +28,8 @@ - - + + @@ -44,7 +44,7 @@ - + @@ -429,7 +429,6 @@
  • PycSphere
  • pycWavelet
  • PYFW
  • -
  • TokemakRec
  • TVDenoiser
  • UncertaintyQuant
  • WaveProp
  • @@ -796,7 +795,7 @@

    Interoperability Protocol @@ -804,7 +803,7 @@

    Interoperability Protocol

    - Created with Sphinx 7.2.5, with some help from ChatGPT (GPT 4). + Created with Sphinx 7.2.6, with some help from ChatGPT (GPT 4).

    diff --git a/fair/plugins/index.html b/fair/plugins/index.html index ed6b9471..65adee10 100644 --- a/fair/plugins/index.html +++ b/fair/plugins/index.html @@ -28,8 +28,8 @@ - - + + @@ -44,7 +44,7 @@ - + @@ -425,7 +425,6 @@
  • PycSphere
  • pycWavelet
  • PYFW
  • -
  • TokemakRec
  • TVDenoiser
  • UncertaintyQuant
  • WaveProp
  • @@ -702,7 +701,7 @@

    pyxu-eigh

    -  

    +  

      @@ -738,7 +737,7 @@

      pyxu-gradient-

      -  

      +  

      @@ -640,44 +593,42 @@ .tooltip:hover .tooltiptext { visibility: visible; } -
      -

      TokemakRec#

      +
      +

      TokamakRec#

      Description#

      -Numquam labore dolore eius quiquia etincidunt non neque. Consectetur quiquia neque sit non labore. Quaerat sed consectetur dolorem labore modi dolor. Etincidunt neque sed aliquam dolore consectetur. Quaerat labore sit labore. Dolore dolore dolor velit non quaerat. - -Porro modi adipisci sed modi adipisci sed. Neque eius non non porro. Quisquam velit modi etincidunt consectetur ipsum. Numquam neque dolor tempora tempora adipisci tempora aliquam. Modi etincidunt ipsum ipsum eius labore eius. Modi non consectetur non dolore aliquam porro. Quisquam eius dolore dolor. Etincidunt eius magnam non dolore. Neque etincidunt eius modi modi est neque. Consectetur sit numquam adipisci porro numquam amet. +Dolorem quiquia quisquam sit neque magnam ipsum. Neque adipisci amet numquam dolore. Ipsum dolor eius aliquam consectetur magnam dolorem aliquam. Eius non est neque dolore quiquia ipsum. Modi eius est ut quaerat sed dolor. Porro tempora quiquia voluptatem. Dolorem dolore sit adipisci adipisci ut sed quisquam. Sit sed numquam dolorem dolorem etincidunt dolore dolorem. -Est numquam quaerat ut sit. Velit velit etincidunt dolor consectetur amet. Sed magnam est neque dolore dolore. Quisquam neque porro neque velit. Dolore est amet consectetur dolore est consectetur. Modi velit quiquia modi quiquia sit dolor porro.
      +Porro quiquia adipisci neque dolore. Modi quaerat velit ipsum quaerat eius sit. Eius neque sed non dolorem. Consectetur est modi ipsum. Dolor amet ut velit. Ipsum quisquam tempora tempora ipsum dolor tempora. Tempora quaerat dolore est. Modi velit quisquam numquam quisquam amet dolorem. Aliquam magnam porro voluptatem non aliquam.

      General information#

      - Short description: Plasma imaging for tokemak data + Short description: Plasma imaging for tokamak data

      -

      +

      - Source code: Go to the source code repository + Source code: Go to the source code repository

      - Documentation: Go to plugin documentation + Documentation: Go to plugin documentation

      Detailed information#

      - Author(s): Michael Smith + Author(s): Christopher Portillo

      - Contact: michael.smith@pyxu_user.org + Contact: christopher.portillo@pyxu_user.org

      @@ -686,7 +637,7 @@

      Detailed information

      Compatibility: - +

      Components contributed#

      @@ -695,19 +646,9 @@

      Components contributed1 - - Solver - 3 - - - - Stop - 2 - - Math - 1 + 2 @@ -719,53 +660,31 @@

      Components contributed(pyxu.math) - -
        - -
      • icky_math - -
      - - Contrib (pyxu.contrib)
        -
      • upbeat_contrib +
      • hot_contrib
      - Operator (pyxu.operator) - -
        - -
      • cooing_operator - -
      - - - Solver (pyxu.solver) + Math (pyxu.math)
        -
      • odd_solver - -
      • sordid_solver +
      • calm_math -
      • light_solver +
      • fine_math
      - Stop (pyxu.stop) + Operator (pyxu.operator)
        -
      • smooth_stop - -
      • square_stop +
      • skinny_operator

      @@ -781,24 +700,6 @@

      Components contributed @@ -830,7 +731,7 @@

      Components contributed diff --git a/fair/plugins_preview/UncertaintyQuant.html b/fair/plugins_preview/UncertaintyQuant.html index 96c0c83d..49164961 100644 --- a/fair/plugins_preview/UncertaintyQuant.html +++ b/fair/plugins_preview/UncertaintyQuant.html @@ -28,8 +28,8 @@ - - + + @@ -44,7 +44,7 @@ - + @@ -429,7 +429,6 @@
    • PycSphere
    • pycWavelet
    • PYFW
    • -
    • TokemakRec
    • TVDenoiser
    • UncertaintyQuant
    • WaveProp
    • @@ -690,7 +689,7 @@

      Detailed information

      Compatibility: - +

      Components contributed#

      @@ -830,7 +829,7 @@

      Components contributed @@ -838,7 +837,7 @@

      Components contributed

      - Created with Sphinx 7.2.5, with some help from ChatGPT (GPT 4). + Created with Sphinx 7.2.6, with some help from ChatGPT (GPT 4).

      diff --git a/fair/plugins_preview/WaveProp.html b/fair/plugins_preview/WaveProp.html index 22427636..643edf03 100644 --- a/fair/plugins_preview/WaveProp.html +++ b/fair/plugins_preview/WaveProp.html @@ -28,8 +28,8 @@ - - + + @@ -44,7 +44,7 @@ - + @@ -428,7 +428,6 @@
    • PycSphere
    • pycWavelet
    • PYFW
    • -
    • TokemakRec
    • TVDenoiser
    • UncertaintyQuant
    • WaveProp
    • @@ -689,7 +688,7 @@

      Detailed information

      Compatibility: - +

      Components contributed#

      @@ -862,7 +861,7 @@

      Components contributed @@ -870,7 +869,7 @@

      Components contributed

      - Created with Sphinx 7.2.5, with some help from ChatGPT (GPT 4). + Created with Sphinx 7.2.6, with some help from ChatGPT (GPT 4).

      diff --git a/fair/plugins_preview/index.html b/fair/plugins_preview/index.html index b5d41798..814688d8 100644 --- a/fair/plugins_preview/index.html +++ b/fair/plugins_preview/index.html @@ -28,8 +28,8 @@ - - + + @@ -44,7 +44,7 @@ - + @@ -425,7 +425,6 @@
    • PycSphere
    • pycWavelet
    • PYFW
    • -
    • TokemakRec
    • TVDenoiser
    • UncertaintyQuant
    • WaveProp
    • @@ -702,7 +701,7 @@

      CSEEG

      -  

      +  

      Components contributed#

      @@ -844,7 +843,7 @@

      Components contributed @@ -852,7 +851,7 @@

      Components contributed

      - Created with Sphinx 7.2.5, with some help from ChatGPT (GPT 4). + Created with Sphinx 7.2.6, with some help from ChatGPT (GPT 4).

      diff --git a/fair/plugins_preview/pycWavelet.html b/fair/plugins_preview/pycWavelet.html index 458488bc..73302248 100644 --- a/fair/plugins_preview/pycWavelet.html +++ b/fair/plugins_preview/pycWavelet.html @@ -28,8 +28,8 @@ - - + + @@ -44,7 +44,7 @@ - + @@ -429,7 +429,6 @@
    • PycSphere
    • pycWavelet
    • PYFW
    • -
    • TokemakRec
    • TVDenoiser
    • UncertaintyQuant
    • WaveProp
    • @@ -644,11 +643,15 @@

      pycWavelet#

      Description#

      -Ipsum consectetur ut sit non dolore. Neque porro quiquia est ut quaerat sit amet. Sed voluptatem ut velit eius eius aliquam dolore. Dolorem ipsum etincidunt non est. Sit quisquam velit eius dolorem modi voluptatem. +Consectetur aliquam quaerat velit porro. Voluptatem ipsum ut sed eius adipisci etincidunt. Ut amet dolor quiquia velit ipsum. Magnam eius dolor sed eius tempora dolor. Voluptatem modi quaerat ipsum aliquam numquam. Adipisci labore adipisci eius. Etincidunt ut amet quiquia sit porro quiquia eius. -Modi quiquia neque quisquam sit etincidunt sit. Amet tempora consectetur etincidunt dolor eius. Aliquam tempora amet neque ipsum. Quiquia porro dolore consectetur. Porro eius non dolor amet magnam non etincidunt. Dolorem dolore quaerat non velit magnam sed aliquam. +Numquam consectetur tempora voluptatem. Dolor tempora numquam dolorem modi magnam eius. Amet quiquia magnam consectetur tempora consectetur numquam. Magnam magnam quaerat consectetur ut numquam porro quisquam. Quisquam tempora dolorem amet tempora adipisci ut. Porro ipsum ut numquam numquam. Dolor non tempora non. -Est labore dolor tempora sit sed quaerat. Quaerat etincidunt quisquam dolore quisquam. Dolor aliquam sit non modi. Amet voluptatem dolorem voluptatem quaerat. Quisquam neque magnam sed modi velit est.
      +Etincidunt dolorem neque dolore velit quaerat. Quisquam ipsum numquam numquam. Dolor neque modi labore non tempora. Sit labore adipisci voluptatem quaerat quisquam ipsum. Neque neque modi adipisci etincidunt. Modi ut neque modi magnam quiquia. Numquam labore eius velit ut ipsum. + +Dolore dolorem ipsum numquam non magnam. Etincidunt quaerat dolore sit quisquam magnam. Porro numquam dolorem magnam labore. Sit velit eius quisquam velit dolore. Quiquia consectetur consectetur eius ut quiquia amet. Dolorem etincidunt ipsum etincidunt non magnam. Labore neque aliquam sed. Ut velit dolore sed. + +Porro dolore magnam non ipsum est quaerat ut. Sed sed dolor adipisci tempora. Dolor est dolor est numquam porro. Velit etincidunt dolorem porro dolorem dolor porro labore. Quisquam dolor ut eius quisquam. Quiquia modi ipsum sed etincidunt.

      General information#

      @@ -658,26 +661,26 @@

      General information -

      +

      - Source code: Go to the source code repository + Source code: Go to the source code repository

      - Documentation: Go to plugin documentation + Documentation: Go to plugin documentation

      Detailed information#

      - Author(s): Cedric Park + Author(s): Ruth Franco

      - Contact: cedric.park@pyxu_user.org + Contact: ruth.franco@pyxu_user.org

      @@ -686,17 +689,17 @@

      Detailed information

      Compatibility: - +

      Components contributed#

      - Solver + Operator 2 - Stop + Solver 3 @@ -705,57 +708,43 @@

      Components contributed3 - - Contrib - 1 - -

      - Math (pyxu.math) + Operator (pyxu.operator)
        -
      • zany_math - -
      • groovy_math +
      • old_operator -
      • stupid_math +
      • dull_operator
      - Stop (pyxu.stop) + Solver (pyxu.solver)
        -
      • nifty_stop +
      • wild_solver -
      • sour_stop +
      • roomy_solver -
      • homely_stop +
      • sulky_solver
      - Solver (pyxu.solver) + Math (pyxu.math)
        -
      • angry_solver +
      • lovely_math -
      • earthy_solver - -
      - - - Contrib (pyxu.contrib) - -
        +
      • tawdry_math -
      • angry_contrib +
      • shrill_math

      @@ -864,7 +853,7 @@

      Components contributed @@ -872,7 +861,7 @@

      Components contributed

      - Created with Sphinx 7.2.5, with some help from ChatGPT (GPT 4). + Created with Sphinx 7.2.6, with some help from ChatGPT (GPT 4).

      diff --git a/fair/score.html b/fair/score.html index 07d3e3c8..f2163bce 100644 --- a/fair/score.html +++ b/fair/score.html @@ -28,8 +28,8 @@ - - + + @@ -44,7 +44,7 @@ - + @@ -429,7 +429,6 @@
    • PycSphere
    • pycWavelet
    • PYFW
    • -
    • TokemakRec
    • TVDenoiser
    • UncertaintyQuant
    • WaveProp
    • @@ -623,7 +622,7 @@

      Contribute to t +
      +
      +
      +_images/LCAV_LOGO.png +
      +
      +
      @@ -1088,7 +1101,7 @@

      Partners & Sponsors @@ -1096,7 +1109,7 @@

      Partners & Sponsors

      - Created with Sphinx 7.2.5, with some help from ChatGPT (GPT 4). + Created with Sphinx 7.2.6, with some help from ChatGPT (GPT 4).

      diff --git a/intro/comparison.html b/intro/comparison.html index bbfd2ea8..c58b1279 100644 --- a/intro/comparison.html +++ b/intro/comparison.html @@ -28,8 +28,8 @@ - - + + @@ -44,7 +44,7 @@ - + @@ -785,7 +785,7 @@

      Conclusion @@ -793,7 +793,7 @@

      Conclusion

      - Created with Sphinx 7.2.5, with some help from ChatGPT (GPT 4). + Created with Sphinx 7.2.6, with some help from ChatGPT (GPT 4).

      diff --git a/intro/index.html b/intro/index.html index 7b13225b..b5dadf7b 100644 --- a/intro/index.html +++ b/intro/index.html @@ -28,8 +28,8 @@ - - + + @@ -44,7 +44,7 @@ - + @@ -628,7 +628,7 @@

      Join Our Community @@ -636,7 +636,7 @@

      Join Our Community

      - Created with Sphinx 7.2.5, with some help from ChatGPT (GPT 4). + Created with Sphinx 7.2.6, with some help from ChatGPT (GPT 4).

      diff --git a/intro/installation.html b/intro/installation.html index 466d1ca7..142d44d2 100644 --- a/intro/installation.html +++ b/intro/installation.html @@ -28,8 +28,8 @@ - - + + @@ -44,7 +44,7 @@ - + @@ -494,41 +494,36 @@

      Basic Installation

      Installation with Optional Dependencies#

      -

      For extended features, you can install optional dependencies:

      - -

      Additionally, you can use aggregate targets based on your platform’s GPU capabilities:

      -
        -
      • complete_no_gpu: full Pyxu install, without GPU support.

      • -
      • complete_gpu: full Pyxu install, with GPU support. (CUDA 12.* required.)

      • -
      -

      Optional dependencies are specified between square brackets:

      -
      pip install pyxu[dev,complete_gpu]
      +

      For extended features, you can install Pyxu with its optional dependencies:

      +
      pip install pyxu[complete-cpu]  # full CPU-only user install.
      +pip install pyxu[complete]      # full CPU/GPU  user install. (CUDA 12.* required.)
       
      +

      More fine-grained extras can be installed by looking at the project.optional-dependencies field in +pyproject.toml at the root of the repository.

      Developer Installation#

      -

      If you’re interested in contributing to Pyxu or experimenting with its codebase, you can clone the repository and -install it manually.

      +

      If you’re interested in contributing to Pyxu or experimenting with its codebase, you must clone the repository and +install it with developer dependencies:

      git clone https://github.com/matthieumeo/pyxu.git
       cd pyxu
      -pip install -e ".[dev,complete_gpu]"  # complete_no_gpu also available
      +pip install -e ".[dev,complete]"
      +pre-commit install
       
      -

      To run tests, you can execute:

      -
      tox run  # optional, see below.
      +

      Pyxu uses tox to automate common operations during the development process. The commands below summarize the main +operations:

      +
      tox run -e py311-test         # run test suite. (fast subset: ~5[min])
      +tox run -e py311-test -- all  # run test suite. (full suite; much longer)
      +tox run -e pre-commit         # run pre-commit hooks.
      +tox run -e doc                # build HTML docs. (incremental update)
      +tox run -e doc -- clean       # build HTML docs. (from scratch)
      +tox run -e dist               # build universal wheels for distribution.
       
      -
      -

      Warning

      -

      Running the full test suite will take several hours. -This step can be skipped.

      -
      -

      For building documentation and running pre-commit hooks:

      -
      tox run -e doc-fast  # build HTML docs
      -tox run -e pre-commit  # run pre-commit hooks
      +

      All available tox environments can be viewed by running:

      +
      tox list
       
      @@ -649,7 +644,7 @@

      You’re All Set! @@ -657,7 +652,7 @@

      You’re All Set!

      - Created with Sphinx 7.2.5, with some help from ChatGPT (GPT 4). + Created with Sphinx 7.2.6, with some help from ChatGPT (GPT 4).

      diff --git a/intro/primer.html b/intro/primer.html index 11763908..29e027ae 100644 --- a/intro/primer.html +++ b/intro/primer.html @@ -28,8 +28,8 @@ - - + + @@ -44,7 +44,7 @@ - + @@ -645,7 +645,7 @@

      Pyxu: High-Performance Computational Imaging with Python @@ -653,7 +653,7 @@

      Pyxu: High-Performance Computational Imaging with Python

      - Created with Sphinx 7.2.5, with some help from ChatGPT (GPT 4). + Created with Sphinx 7.2.6, with some help from ChatGPT (GPT 4).

      diff --git a/intro/tomo.html b/intro/tomo.html index 7cb2bec5..68d76016 100644 --- a/intro/tomo.html +++ b/intro/tomo.html @@ -18,8 +18,8 @@ - - + + @@ -28,12 +28,12 @@ - + - + @@ -340,7 +340,7 @@

      A Non-Ideal Parallel-beam Tomographic Setup

      Naive Application of Filtered-Back Projection#

      -

      In this section, we explore what happens when we naively apply the iradon() function from the scikit-image library, which implements the Filtered Back Projection (FBP) algorithm, without accounting for the non-idealities inherent to our tomographic setup. Despite the algorithm’s effectiveness in idealized scenarios, we observe that our reconstructed image is riddled with severe artifacts.

      +

      In this section, we explore what happens when we naively apply iradon()🔗 from scikit-image, which implements the Filtered Back Projection (FBP) algorithm, without accounting for the non-idealities inherent to our tomographic setup. Despite the algorithm’s effectiveness in idealized scenarios, we observe that our reconstructed image is riddled with severe artifacts.

      We attempt to mitigate these artifacts by filling in the gaps in the sinogram through matched filtering. However, as can be seen below, this approach proves insufficient for handling the complex distortions caused by the non-ideal conditions in our setup. This section serves as a poignant illustration of the limitations of using off-the-shelf Radon transform methods in scenarios that deviate from their underlying assumptions.

      [4]:
      @@ -383,7 +383,8 @@ 

      Naive Application of Filtered-Back Projection

      Pyxu’s Modularity to the Rescue#

      -

      We now turn our attention to how Pyxu addresses the limitations encountered when using iradon() for reconstruction in non-ideal conditions. Unlike monolithic approaches, Pyxu’s microservice architecture thrives on modularity, allowing us to decompose the data model into a sequence of simple linear operators: the Radon transform (\(R\)), followed by filtering (\(F\)), masking (\(M\)), and finally tapering (\(T\)).

      +

      We now turn our attention to how Pyxu addresses the limitations encountered when using iradon()🔗 for reconstruction in non-ideal conditions. Unlike monolithic approaches, Pyxu’s microservice architecture thrives on modularity, allowing us to decompose the data model into a sequence of simple linear operators: the Radon transform (\(R\)), followed by filtering (\(F\)), masking +(\(M\)), and finally tapering (\(T\)).

      \[\Phi= T \circ M \circ F \circ R\]

      Here is a code snippet detailing the construction of such an operator with Pyxu:

      @@ -427,8 +428,9 @@

      Pyxu’s Modularity to the Rescueapply and adjoint methods of the composite operator \(\Phi\) are automatically inferred by Pyxu, leveraging standard composition rules.

      In Pyxu, one of the powerful features is the ability to define matrix-free linear operators, a crucial advantage for large-scale imaging problems. In layman’s terms, you define an operator by just coding its apply and adjoint methods. This bypasses the need for storing the operator as a cumbersome matrix, which not only saves memory but also boosts computational speed when the matrix is structured.

      -

      In the example above, we have for example used the radon() and iradon() functions from scikit-image to define the apply and adjoint methods of our Radon operator. This exemplifies Pyxu’s core philosophy: you don’t have to reinvent the wheel; leverage useful components from other libraries when available. Pyxu is designed to stitch these disparate elements together into a seamless, modular workflow.

      -

      Internally, Pyxu maintains a detailed record of the composition chains and arithmetic operations that define each operator. This comprehensive tracking is made accessible through the operator’s expr() method, which can reveal the intricate web of calculations and transformations that the operator encompasses.

      +

      In the example above, we have for example used the radon()🔗 and iradon()🔗 functions from scikit-image to define the apply and adjoint methods of our Radon operator. This exemplifies Pyxu’s core philosophy: you don’t have to reinvent the wheel; leverage useful components from other +libraries when available. Pyxu is designed to stitch these disparate elements together into a seamless, modular workflow.

      +

      Internally, Pyxu maintains a detailed record of the composition chains and arithmetic operations that define each operator. This comprehensive tracking is made accessible through the operator’s expr()🔗 method, which can reveal the intricate web of calculations and transformations that the operator encompasses.

      [6]:
       
      @@ -497,7 +499,7 @@

      Image Reconstruction with Pyxu#

      Now that we’ve accurately modeled our acquisition system using the composite operator \(\Phi\), we can proceed to the reconstruction phase. Traditional methods like filtered backprojection are no longer applicable here due to the complexities introduced in our custom setup. Consequently, we must resort to generic solvers such as the conjugate gradient method to solve the corresponding (dampened) normal equations.

      Forward model

      -

      In Pyxu, this task is greatly simplified by the pinv() (pseudoinverse) method provided by LinOp instances. The pinv() method allows us to implement simple reconstruction techniques without getting lost in the details.

      +

      In Pyxu, this task is greatly simplified by the pinv()🔗 (pseudoinverse) method provided by LinOp🔗 instances. The pinv() method allows us to implement simple reconstruction techniques without getting lost in the details.

      The following code snippet illustrates its application in our tomographic setting. For the sake of this demonstration, we employ a custom stopping criterion (relative error smaller than \(10^{-3}\) and 500 iterations max) and incorporate a dampening factor of 4 to improve the algorithm’s resilience to noise and numerical instability.

      [9]:
      @@ -607,9 +609,9 @@ 

      Bayesian Inversion -
    • Gradient(): Computes the gradient of an image. The argument arg_shape=phantom.shape specifies the shape of the image, i.e., phantom.shape.

    • +
    • Gradient()🔗: Computes the gradient of an image. The argument arg_shape=phantom.shape specifies the shape of the image, i.e., phantom.shape.

    • lambda_: Scales the total variation term. It is calculated based on the noise level sigma in the observations.

    • -
    • moreau_envelope(0.01): Applies the Moreau envelope to smooth the TV term, facilitating the downstream MAP estimation process.

    • +
    • moreau_envelope(0.01)🔗: Applies the Moreau envelope to smooth the TV term, facilitating the downstream MAP estimation process.

    Defining the Positivity and L1 Norm Constraints

    # Positivity + L1 norm
    @@ -617,7 +619,7 @@ 

    Bayesian Inversion -
  • PositiveL1Norm(): This operator combines a sparsity-promoting \(L_1\) norm and a positivity constraint.

  • +
  • PositiveL1Norm()🔗: This operator combines a sparsity-promoting \(L_1\) norm and a positivity constraint.

  • Defining the Data Likelihood (Loss)

    # Loss
    @@ -625,8 +627,8 @@ 

    Bayesian Inversion -
  • SquaredL2Norm(): Represents the squared L2 norm of the residual between the observed and modeled data, scaled by the noise level sigma.

  • -
  • .asloss(sino.ravel()): Sets the observed sinogram as the data.

  • +
  • SquaredL2Norm()🔗: Represents the squared L2 norm of the residual between the observed and modeled data, scaled by the noise level sigma.

  • +
  • .asloss(sino.ravel())🔗: Sets the observed sinogram as the data.

  • * Phi: Compose the \(L_2\) norm with \(\Phi\), the operator modeling the acquisition process.

  • Defining the Smooth Part of the Posterior

    @@ -637,7 +639,7 @@

    Bayesian Inversion
  • smooth_posterior: The smooth part of the posterior that combines both the likelihood (loss) and the smoothed tv_prior. Will be useful later on.

  • -
  • estimate_diff_lipschitz() allows to compute the Lipschitz constant of the gradient of the differentiable part of the posterior. This value will be used later on to optimally set the step size of the gradient-based optimizer. This step can be time consuming for large scale problems.

  • +
  • estimate_diff_lipschitz()🔗 allows to compute the Lipschitz constant of the gradient of the differentiable part of the posterior. This value will be used later on to optimally set the step size of the gradient-based optimizer. This step can be time consuming for large scale problems.

  • We are now ready to solve the MAP estimation problem. We do that via proximal gradient descent (PGD), a generic solver for problems of the form

    By understanding and applying these principles, one can utilize PnP schemes to achieve high-quality image reconstructions that are both data-consistent and regularized, all while benefiting from the computational and modeling advantages that come with implicit priors.

    -

    In Pyxu, implicit PnP priors can be implemented in both forms via the ProxDiffFunc() base class. Here is a simple example with a median filter denoiser.

    +

    In Pyxu, implicit PnP priors can be implemented in both forms via the ProxDiffFunc()🔗 base class. Here is a simple example with a median filter denoiser.

    diff --git a/intro/tomo.ipynb b/intro/tomo.ipynb index 9acb13fd..bb5b61e2 100644 --- a/intro/tomo.ipynb +++ b/intro/tomo.ipynb @@ -216,10 +216,11 @@ "source": [ "## Naive Application of Filtered-Back Projection\n", "\n", - "In this section, we explore what happens when we naively apply the `iradon()` function from the `scikit-image` library,\n", - "which implements the *Filtered Back Projection (FBP)* algorithm, without accounting for the non-idealities inherent to\n", - "our tomographic setup. Despite the algorithm's effectiveness in idealized scenarios, we observe that our reconstructed\n", - "image is riddled with severe artifacts.\n", + "In this section, we explore what happens when we naively apply\n", + "`iradon()`[🔗](https://scikit-image.org/docs/stable/api/skimage.transform.html#skimage.transform.iradon) from\n", + "`scikit-image`, which implements the *Filtered Back Projection (FBP)* algorithm, without accounting for the\n", + "non-idealities inherent to our tomographic setup. Despite the algorithm's effectiveness in idealized scenarios, we\n", + "observe that our reconstructed image is riddled with severe artifacts.\n", "\n", "We attempt to mitigate these artifacts by filling in the gaps in the sinogram through matched filtering. However, as can\n", "be seen below, this approach proves insufficient for handling the complex distortions caused by the non-ideal conditions\n", @@ -296,10 +297,11 @@ "source": [ "## Pyxu's Modularity to the Rescue\n", "\n", - "We now turn our attention to how Pyxu addresses the limitations encountered when using `iradon()` for reconstruction in\n", - "non-ideal conditions. Unlike monolithic approaches, Pyxu’s microservice architecture thrives on modularity, allowing us\n", - "to decompose the data model into a sequence of simple linear operators: the Radon transform ($R$), followed by filtering\n", - "($F$), masking ($M$), and finally tapering ($T$).\n", + "We now turn our attention to how Pyxu addresses the limitations encountered when using\n", + "`iradon()`[🔗](https://scikit-image.org/docs/stable/api/skimage.transform.html#skimage.transform.iradon) for\n", + "reconstruction in non-ideal conditions. Unlike monolithic approaches, Pyxu’s microservice architecture thrives on\n", + "modularity, allowing us to decompose the data model into a sequence of simple linear operators: the Radon transform\n", + "($R$), followed by filtering ($F$), masking ($M$), and finally tapering ($T$).\n", "\n", "$$\\Phi= T \\circ M \\circ F \\circ R$$\n", "\n", @@ -381,10 +383,12 @@ "methods. This bypasses the need for storing the operator as a cumbersome matrix, which not only saves memory but also\n", "boosts computational speed when the matrix is structured.\n", "\n", - "In the example above, we have for example used the `radon()` and `iradon()` functions from `scikit-image` to define the\n", - "`apply` and `adjoint` methods of our Radon operator. This exemplifies Pyxu's core philosophy: you don't have to reinvent\n", - "the wheel; leverage useful components from other libraries when available. Pyxu is designed to stitch these disparate\n", - "elements together into a seamless, modular workflow." + "In the example above, we have for example used the\n", + "`radon()`[🔗](https://scikit-image.org/docs/stable/api/skimage.transform.html#skimage.transform.radon) and\n", + "`iradon()`[🔗](https://scikit-image.org/docs/stable/api/skimage.transform.html#skimage.transform.iradon) functions from\n", + "`scikit-image` to define the `apply` and `adjoint` methods of our Radon operator. This exemplifies Pyxu's core\n", + "philosophy: you don't have to reinvent the wheel; leverage useful components from other libraries when available. Pyxu\n", + "is designed to stitch these disparate elements together into a seamless, modular workflow." ] }, { @@ -399,8 +403,9 @@ }, "source": [ "Internally, Pyxu maintains a detailed record of the composition chains and arithmetic operations that define each\n", - "operator. This comprehensive tracking is made accessible through the operator's `expr()` method, which can reveal the\n", - "intricate web of calculations and transformations that the operator encompasses." + "operator. This comprehensive tracking is made accessible through the operator's\n", + "`expr()`[🔗](../api/abc.html#pyxu.abc.Operator.expr) method, which can reveal the intricate web of calculations and\n", + "transformations that the operator encompasses." ] }, { @@ -589,8 +594,9 @@ "tags": [] }, "source": [ - "In Pyxu, this task is greatly simplified by the `pinv()` (pseudoinverse) method provided by `LinOp` instances. The\n", - "`pinv()` method allows us to implement simple reconstruction techniques without getting lost in the details. \n", + "In Pyxu, this task is greatly simplified by the `pinv()`[🔗](../api/abc.html#pyxu.abc.LinOp.pinv) (pseudoinverse) method\n", + "provided by `LinOp`[🔗](../api/abc.html#pyxu.abc.LinOp) instances. The `pinv()` method allows us to implement simple\n", + "reconstruction techniques without getting lost in the details. \n", "\n", "The following code snippet illustrates its application in our tomographic setting. For the sake of this demonstration,\n", "we employ a custom stopping criterion (relative error smaller than $10^{-3}$ and 500 iterations max) and incorporate a\n", @@ -820,27 +826,28 @@ "huber_norm = L1Norm(grad.shape[0]).moreau_envelope(0.01)\n", "tv_prior = lambda_ * huber_norm * grad\n", "```\n", - "- `Gradient()`: Computes the gradient of an image. The argument `arg_shape=phantom.shape` specifies the shape of the\n", - " image, i.e., `phantom.shape`.\n", + "- `Gradient()`[🔗](../api/operator/linop.html#pyxu.operator.Gradient): Computes the gradient of an image. The argument\n", + " `arg_shape=phantom.shape` specifies the shape of the image, i.e., `phantom.shape`.\n", "- `lambda_`: Scales the total variation term. It is calculated based on the noise level `sigma` in the observations.\n", - "- `moreau_envelope(0.01)`: Applies the Moreau envelope to smooth the TV term, facilitating the downstream MAP estimation\n", - " process.\n", + "- `moreau_envelope(0.01)`[🔗](../api/abc.html#pyxu.abc.ProxFunc.moreau_envelope): Applies the Moreau envelope to smooth\n", + " the TV term, facilitating the downstream MAP estimation process.\n", "\n", "**Defining the Positivity and L1 Norm Constraints**\n", "```python\n", "# Positivity + L1 norm \n", "posL1 = 0.05 * PositiveL1Norm(phantom.size)\n", "```\n", - "- `PositiveL1Norm()`: This operator combines a sparsity-promoting $L_1$ norm and a positivity constraint.\n", + "- `PositiveL1Norm()`[🔗](../api/operator/func.html#pyxu.operator.PositiveL1Norm): This operator combines a\n", + " sparsity-promoting $L_1$ norm and a positivity constraint.\n", "\n", "**Defining the Data Likelihood (Loss)**\n", "```python\n", "# Loss\n", "loss = (1/ (2 * sigma**2)) * SquaredL2Norm(dim=sino.size).asloss(sino.ravel()) * Phi\n", "```\n", - "- `SquaredL2Norm()`: Represents the squared L2 norm of the residual between the observed and modeled data, scaled by the\n", - " noise level `sigma`.\n", - "- `.asloss(sino.ravel())`: Sets the observed sinogram as the data.\n", + "- `SquaredL2Norm()`[🔗](../api/operator/func.html#pyxu.operator.SquaredL2Norm): Represents the squared L2 norm of the\n", + " residual between the observed and modeled data, scaled by the noise level `sigma`.\n", + "- `.asloss(sino.ravel())`[🔗](../api/abc.html#pyxu.abc.Func.asloss): Sets the observed sinogram as the data.\n", "- `* Phi`: Compose the $L_2$ norm with $\\Phi$, the operator modeling the acquisition process.\n", "\n", "**Defining the Smooth Part of the Posterior**\n", @@ -851,9 +858,9 @@ "```\n", "- `smooth_posterior`: The smooth part of the posterior that combines both the likelihood (loss) and the smoothed\n", " `tv_prior`. Will be useful later on.\n", - "- `estimate_diff_lipschitz()` allows to compute the Lipschitz constant of the gradient of the differentiable part of the\n", - " posterior. This value will be used later on to optimally set the step size of the gradient-based optimizer. This step\n", - " can be time consuming for large scale problems." + "- `estimate_diff_lipschitz()`[🔗](../api/abc.html#pyxu.abc.DiffMap.estimate_diff_lipschitz) allows to compute the\n", + " Lipschitz constant of the gradient of the differentiable part of the posterior. This value will be used later on to\n", + " optimally set the step size of the gradient-based optimizer. This step can be time consuming for large scale problems." ] }, { @@ -1061,8 +1068,9 @@ "id": "69adcacb-298b-47ff-af12-a5e60f84d879", "metadata": {}, "source": [ - "In Pyxu, implicit PnP priors can be implemented in both forms via the `ProxDiffFunc()` base class. Here is a simple\n", - "example with a median filter denoiser. " + "In Pyxu, implicit PnP priors can be implemented in both forms via the\n", + "`ProxDiffFunc()`[🔗](../api/abc.html#pyxu.abc.ProxDiffFunc) base class. Here is a simple example with a median filter\n", + "denoiser." ] }, { diff --git a/objects.inv b/objects.inv index d546bf7581bf1e1ecb3d2c1999fad40f440466cf..4c15b3cc4a25b9f25669a3a794cf9f3557c24020 100644 GIT binary patch delta 17484 zcmV)m69L+?CHK)60 zB%R(&WM@`&=b5_XsjTWzuQE_1GLutGlEX_?<*c;SLbevR!(nT24MVt~l81_|^z~PDUZDwJ5eh0@*Ewle1epevxAjOh;}=iTe;MQ#kQPq_idgYHuh<=PFa5zCC^q4#dG+K;=;ts#l@|t9_`&wzAH$<#V&sPXj@wQ72xVu z`vhqi9l*u5c5{Dn!mn9wt8b4ATNmqkReKhDlwtdH6~9y%{|bK0dAVQZ>Ce!5yh@7& z-h!=dkYy^Ztk37Oo6W`N*%fs|Wtm5hRf%D)i{<6z#ofit{c`r<>S8j$=JMfYe!slE zaX^x;PU*N>-dxP?oE~XvyE#j+%Hs*UsbBwADPM&d`X4uxXF|}ndY#74IJdF2O_TIOMM=iN@WL$p4sL#r zRdKXa7@`twqZr;%OFz6tRa|b0GR-ytsd;v^TU&=X-MPp-u*YUcbs&ktnVnDi_%ja5 z=oxReqSC;9i!`pvC`~NANC7EqU2taYl3M1(OPYT_g;8$%pk7E3!jC9;vSPbVLu^xx zsr~tE{pDtETeXK=6)11#m z`2%ou-Ohs;quiDBJW95yv$I9#Q7`?@2VKv`{lbeovu1N(>be?^L&q@ z#L0himA>HI$oYsQfAPlwCp#YT_ zRrFS}jS$k(fO4C`FmCnffaQcK7y65MMu^%}htpr)Gvd^yS{%)Cgc+$Z)?xLRDvU_A zsTK#SXGlwLabpODn;LLlE$UIr+=OR=WyF6mOnHo8ZF?P}VOC;9Y};!QdqV;vRJ&s zN<-RT{FdSAj5q-|^Z#rJ?)>`8!_}d{zME>I z=MSqdn~NK|cE6w~ba7A=zz&AS`DT-j1sMbLaD9`i1uYD>zHocoUBK0s#q55Q;{`-T zb^W+|HCRE>xi9Vof483+@mrEUT6#K6Tm z4Oed``%X>z+et}(>n@+nbExp+Bz_6vE`xO`?!CeEF6labCS&XwDavT)uH)vxxo=^v z6T1e15yxqudoOGSGNFeN2t|<6_fN0-&wp&KpvZ)L6YQ(xX%b9GMvh5JU$SkoIGvWs z2VCqY0g$;eIuz3&NFp{*I8Sgq2}i#<+y;Rz$h?`1@OCo7J6D7%j^$=9w~9{yTs!d< zKb67B@tkgST8qzL+~!jh<)?o`3Je zV0sg_M7v<8MMT{vjAJAO-71O_x5brJj$fkEeQbY+lAZI)j&zb2 z$r2_dW9VI#(^jg;P#Nqs2cpYl6~Wt)btK|gk$6lFO*LYGs3NTO9AC;Tt_&JvqmUW7hr_1KSP8P z)t*`gdN8BmeFl`E<*g;5n|~vEuNt86!fychtt;}|acEcMF<|}TKN!E|Zl*CUkJuK{ z(PvQEC*3)=3lo|Yc9!i%9$D_Px1)%5hct5C9ZS3uXIJ-B zNA9G~ix5S+Ef0ge?3i~JSFUsM;uq)Thk}hHU7e>56x|n$Cae}cyNWL5DI&AQ^>u%x zw2n6Wh3MeoH~;A5wbG^$;P_d;d<*d0qr+nw$9QF0lLBC*Nq>mS&y`D(7k7@Ni!ZK8 zUZCfw_!1>fa?@%@T<%b?-==x!sQvPTtMI;PM zb}Hr4&zd=}k}^8<4~P)q`j2E~-^yeMRl7JvQ;el0g!RFs<-Cl-Fq;x~UnVjyAd z9irlC^Q(53lF6B$6V-g<1au?aVYo>XzVQ$26S8m9EnLcUQySl_*zi3>%Cq3FxI84l>lsItF6L1!VN@4hI41ogV|S z;{qJGjkIt^dVh#LDZh2Z7&lJsTKOjDZwdc-sYcpwMoW>*RsB&F%C3Km$umraD07%^g4O05dYIHI=visDD3) zFzYX|pxDZ%Te{$3E^)yS?GcXxq_5k9}lJy_S(=vq7dzNqvJ^hQrK%l2Z=N{ zw}VNwx4w^942og9QZ1jR;$w^%B(-~@y5~9iG_xpqhEp(WP1qns@f<#*xPY@r2Yv7$ zmqoq4@8aJNBd21AIGCXJFoYZEfPejetM51W`?M?DY{+0P-fV!djlDK>fQWxp31pF< zV0MEB@ZDO3Yc$**=;H@uShE{QAzaFx_dqhm!w_zuOW3V+uoA9ZcGn3pfCHB}+Sl6k zLPo$z8AV|TE9kdfA7jiosdyJ@$6CL@G)0Z`9;(=ymiriE#z_;rvnJ&}#($V`(ky+k zCgncHm~qmgdbB3xKE{|aQn{XG#acU=qaeI=g~^(iIhmtIc?(#3bK7#z2aj z$he0ge2mVPOiu+M!mpWCv&tRx!DC!|R9I0_du`|#kzK!ct8p8O9AoD%?rezVt3GY` zDBUj=#=lw;3Lj&Pi8L?wB!44*2G%?E64_pufg($b5N}}w%FVK!5ED39dsp|cX1Rd_ z1}v}l=!A@bE9;sYzVD;3;nTT8jK)+2DA`~#wX)JOpWSROKC^q|O;KaK^2EF{LXWU6 zuclZmFE20dE^h9ZGrs+2*uYN7rhYH!E@^Z5a5KMO-rjg%%Q2Bsm4D@flJMF`)kKc5 z)B7P%ZpD1Hyt$a&d8P;$<0cC6AxldPTeItMY?h0Q>(%W&$!rf`1C+#PN-lD4Ztt$O z526(hl#z*32R5X&WoJ;jC|cdFm-oxh7n{Y!wh#CNq=~l-3??Fj|a-A$g@NF(M87tc?`3=FW#8Jn39Zt4Px62 zukLRDW_fLBf}6-O_Fv=`=2qk+ix+pZ`{mplh%-~a+PpRFTo$hh9U!tH=@S6J&$l=U z_Bh_KBRDOX+K$7~V;T-=j#0J2$uo}A3|qCk@0 ztzZb3)q+zCU9;+>kB7;iKQ{$Ekk4L7<2!0$5JP(v(W$`Z18d?i;|}VypcQQ5g6yH< zXq`P&>!CeV;yWOl=eh2n}HR&lodK%1>;yBmHo7JO9x$#*C95 z*cZT-tk<)f2X!r^7iHpQvvy!zV79H{o(LHM7dMYuP}FyoYQs%*q&{X*vZMVi&Y=yY zaGKXn=6@(C?|Bddm-l)U!gb65tc2cj>n4PGj>LmyfScPMO{eA6#v5eNr63sX1F=-g zHUG_F@oPqy4mc@Me9k`z%k`hpXf2!XYIM=V6!If#qq~KMaV%GnM!)mJv5Jw6Ml;m) zmxrsFHxfTh=)ly`S#f7>2{3d{Lz>VPP=3b=4u22nLA%KBl9F47JQv8<9NEtNq^UZu zvQRq%K2_o*fs@z7q&`)7nHEuT%GCWRF`8Xz=x3BiC<#tlj6O?DsG)NjPS$%5_uW%^?$h_%z+U?# zeSbDpKenReng-p16VP}hJvlz0{K={@kuG=qUrrFwJBai{X&%MK&a>5$xQW{!JC0}M zm!HeXA()QkFlfE$$EROm7Ji(*_%Wt;dkr#}-Db^)?DWga!uz_8`=-yHqoYsPU2qcm z>`FHrju7iSX6##;4$`w6mx_08T-4r;4QW$(b zsNrTxaR!5*REI~L`+Jmxq#fK1Qks0`;v|S07Vdo#;a$q+GST6bsdmjt>Ae)-7}M(G zzeEKF=9$6eLJ2nK7(d;7m_XWG@_%~}^-ii07{%_g`MV3Z!wJzkT+^h@NB(aYB7X|s z4!YBLnIoo(5rywYFy49BQ4#d~n+v zfCZ~(e8qRI84D9VJZBJ*8R&A2?g69iWl;dLP3u6vnkl^BcvtkxTVvMA^njYyp7Cub zpo^Xz`P~>ppweb_yr=*6Axy`uw0|(P2d;Eb)DGDC*!$o%qZh?c>9@<-D3}mJ8D4jk{**Z+8IrG&FUBdJW%hs`&T7OV^;^io2 zb2Mgiw44@^9?vum$E@mOpSnOr5M{?f5N@#!qSdkI%(4*F2zE@@t)D$yeyK*`2Wj#w zUM>%}0(uW|IR|ae=Sp~&u4I>~_*tHYRZtESb#28(EGzN52Tc!4*|I)m5VbM8um6-- zqAfE`a%6P#lAzc8DG044{C_%T4AA%!bm%J(;R?BU12`}glNO!9Sm)JY8>+vEKk%lq!(sM?lK1S+QB|hwY96|O z%0fyXX-1Hi66q(CFs&I_t=JkhNvGIIWmecW0BZ}gKNw^gcOJDc%Gpo7GG#u^z9$hJORpM+MFq4&h&_H~^aoGN)vyR*NZfhmEJC zK!4z^qx$ABT!0}=_5MiyDu_4-k)NpVn)({TEhS8O+%>1^5%&#X>Ym~-ddB%q|5f;? 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dim: int): - super().__init__(shape=(1, dim)) - def apply(self, arr): - return np.linalg.norm(arr, axis=-1, keepdims=True, ord=1) - def prox(self, arr, tau): - return np.clip(np.abs(arr)-tau, a_min=0, a_max=None) * np.sign(arr) - -N = 512 -l1_norm = L1Norm(dim=N) -mus = [0.1, 0.5, 1] -smooth_l1_norms = [l1_norm.moreau_envelope(mu) for mu in mus] - -x = np.linspace(-1, 1, N)[:, None] -labels=['mu=0'] -labels.extend([f'mu={mu}' for mu in mus]) -plt.figure() -plt.plot(x, l1_norm(x)) -for f in smooth_l1_norms: - plt.plot(x, f(x)) -plt.legend(labels) -plt.title('Moreau Envelope') - -labels=[f'mu={mu}' for mu in mus] -plt.figure() -for f in smooth_l1_norms: - plt.plot(x, f.grad(x)) -plt.legend(labels) -plt.title('Derivative of Moreau Envelope') \ No newline at end of file diff --git a/plot_directive/api/abc/operator-1_00.hires.png b/plot_directive/api/abc/operator-1_00.hires.png deleted file mode 100644 index b5e99c7d933d8574967c81bc1c582c43a849028d..0000000000000000000000000000000000000000 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a/plot_directive/api/util/misc-2.py b/plot_directive/api/util/misc-2.py deleted file mode 100644 index 2ebee297..00000000 --- a/plot_directive/api/util/misc-2.py +++ /dev/null @@ -1,7 +0,0 @@ -import numpy as np -import matplotlib.pyplot as plt -from pyxu.util.misc import star_like_sample - -star = star_like_sample(N=256, w=8, s=20, po=3, x0=0.7) -plt.figure() -plt.imshow(star) \ No newline at end of file diff --git a/py-modindex.html b/py-modindex.html index eac6bdab..9d373c74 100644 --- a/py-modindex.html +++ b/py-modindex.html @@ -27,8 +27,8 @@ - - + + @@ -43,7 +43,7 @@ - + @@ -523,7 +523,7 @@

    Python Module Index

    diff --git a/search.html b/search.html index 19b10c83..4c26b905 100644 --- a/search.html +++ b/search.html @@ -26,8 +26,8 @@ - - + + @@ -42,7 +42,7 @@ - + @@ -502,7 +502,7 @@

    Search

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