Simplify implementation of periodic kernels.

dev_requirements.txt

@@ -502,10 +502,6 @@ more-itertools==9.0.0
     #   -r gptools-torch/test_requirements.txt
     #   -r gptools-util/test_requirements.txt
     #   jaraco-classes
-mpmath==1.2.1
-    # via
-    #   -r doc_requirements.txt
-    #   -r gptools-util/test_requirements.txt
 myst-nb==0.17.1
     # via -r doc_requirements.txt
 myst-parser==0.18.1

doc_requirements.txt

@@ -224,8 +224,6 @@ mistune==2.0.4
     # via nbconvert
 more-itertools==9.0.0
     # via jaraco-classes
-mpmath==1.2.1
-    # via gp-tools-util
 myst-nb==0.17.1
     # via -r doc_requirements.in
 myst-parser==0.18.1

gptools-stan/docs/poisson_regression/poisson_regression.md

@@ -236,9 +236,7 @@ Given the substantial performance improvements, we can readily increase the samp
 
 ```{code-cell} ipython3
 x = np.arange(1024)
-num_terms = 3 if os.environ.get("CI") else None
-kernel = ExpQuadKernel(sigma=1.2, length_scale=15, period=x.size, num_terms=num_terms) \
-    + DiagonalKernel(1e-3, x.size)
+kernel = ExpQuadKernel(sigma=1.2, length_scale=15, period=x.size) + DiagonalKernel(1e-3, x.size)
 sample = simulate(x, kernel)
 plot_sample(sample).legend()
 ```

gptools-stan/docs/poisson_regression/poisson_regression_centered.stan

@@ -21,7 +21,7 @@ parameters {
 model {
     // Gaussian process prior and observation model.
     matrix[n, n] cov = add_diag(gp_periodic_exp_quad_cov(X, X, sigma, rep_vector(length_scale, 1),
-                                                         rep_vector(n, 1), 10), epsilon);
+                                                         rep_vector(n, 1)), epsilon);
     eta ~ multi_normal(zeros_vector(n), cov);
     y ~ poisson_log(eta);
 }

gptools-stan/docs/poisson_regression/poisson_regression_fourier_centered.stan

@@ -16,7 +16,7 @@ parameters {
 
 transformed parameters {
     // Evaluate covariance of the point at zero with everything else.
-    vector[n %/% 2 + 1] cov_rfft = gp_periodic_exp_quad_cov_rfft(n, sigma, length_scale, n, 10)
+    vector[n %/% 2 + 1] cov_rfft = gp_periodic_exp_quad_cov_rfft(n, sigma, length_scale, n)
         + epsilon;
 }
 

gptools-stan/docs/poisson_regression/poisson_regression_fourier_non_centered.stan

@@ -16,7 +16,7 @@ parameters {
 
 transformed parameters {
     vector[n] eta;
-    vector[n %/% 2 + 1] cov_rfft = gp_periodic_exp_quad_cov_rfft(n, sigma, length_scale, n, 10)
+    vector[n %/% 2 + 1] cov_rfft = gp_periodic_exp_quad_cov_rfft(n, sigma, length_scale, n)
         + epsilon;
     eta = gp_transform_inv_rfft(z, zeros_vector(n), cov_rfft);
 }

gptools-stan/docs/poisson_regression/poisson_regression_non_centered.stan

@@ -20,7 +20,7 @@ transformed parameters {
         // wrap the evaluation in braces because Stan only writes top-level variables to the output
         // CSV files, and we don't need to store the entire covariance matrix.
         matrix[n, n] cov = add_diag(gp_periodic_exp_quad_cov(
-            X, X, sigma, rep_vector(length_scale, 1), rep_vector(n, 1), 10), epsilon);
+            X, X, sigma, rep_vector(length_scale, 1), rep_vector(n, 1)), epsilon);
         eta = cholesky_decompose(cov) * z;
     }
 }

gptools-stan/gptools/stan/gptools_kernels.stan

@@ -80,39 +80,27 @@ matrix dist2(array [] vector x, vector period, vector scale) {
 Evaluate the periodic squared exponential kernel.
 */
 matrix gp_periodic_exp_quad_cov(array [] vector x1, array [] vector x2, real sigma,
-                                vector length_scale, vector period, int nterms) {
+                                vector length_scale, vector period) {
     int m = size(x1);
     int n = size(x2);
-    matrix[m, n] result;
-    vector[size(length_scale)] time = 2 * (pi() * length_scale ./ period) ^ 2;
-    vector[size(length_scale)] q = exp(-time);
-    real scale = sigma * sigma * prod(sqrt(time / pi()));
-    for (i in 1:m) {
-        for (j in 1:n) {
-            result[i, j] = scale * prod(jtheta(evaluate_residuals(x1[i], x2[j], period, period), q, nterms));
-        }
-    }
-    return result;
+    return sigma * sigma * exp(-dist2(x1, x2, period, length_scale) / 2);
 }
 
 /**
 Evaluate the real fast Fourier transform of the periodic squared exponential kernel.
 */
-vector gp_periodic_exp_quad_cov_rfft(int n, real sigma, real length_scale, real period,
-                                     int nterms) {
-    real time = 2 * (pi() * length_scale / period) ^ 2;
-    return sigma * sigma * jtheta_rfft(n, exp(-time), nterms) * sqrt(time / pi());
+vector gp_periodic_exp_quad_cov_rfft(int n, real sigma, real length_scale, real period) {
+    int nrfft = n %/% 2 + 1;
+    return n * sigma ^ 2 * length_scale / period * sqrt(2 * pi())
+        * exp(-2 * (pi() * linspaced_vector(nrfft, 0, nrfft - 1) * length_scale / period) ^ 2);
 }
 
 /**
 Evaluate the two-dimensional real fast Fourier transform of the periodic squared exponential kernel.
 */
-matrix gp_periodic_exp_quad_cov_rfft2(int m, int n, real sigma, vector length_scale, vector period,
-                                      int nterms) {
-    vector[m %/% 2 + 1] rfftm = gp_periodic_exp_quad_cov_rfft(m, sigma, length_scale[1], period[1],
-                                                              nterms);
-    vector[n %/% 2 + 1] rfftn = gp_periodic_exp_quad_cov_rfft(n, 1, length_scale[2], period[2],
-                                                              nterms);
+matrix gp_periodic_exp_quad_cov_rfft2(int m, int n, real sigma, vector length_scale, vector period) {
+    vector[m %/% 2 + 1] rfftm = gp_periodic_exp_quad_cov_rfft(m, sigma, length_scale[1], period[1]);
+    vector[n %/% 2 + 1] rfftn = gp_periodic_exp_quad_cov_rfft(n, 1, length_scale[2], period[2]);
     return get_real(expand_rfft(rfftm, m)) * rfftn';
 }
 

gptools-stan/gptools/stan/gptools_util.stan

@@ -425,23 +425,3 @@ real std_normal_lpdf(complex_vector z) {
 real std_normal_lpdf(matrix z) {
     return std_normal_lpdf(to_vector(z));
 }
-
-// Special functions -------------------------------------------------------------------------------
-
-vector jtheta(vector z, vector q, int nterms) {
-    vector[size(z)] result = zeros_vector(size(z));
-    for (n in 1:nterms) {
-        result += q ^ (n ^ 2) .* cos(2 * pi() * z * n);
-    }
-    return 1 + 2 * result;
-}
-
-vector jtheta_rfft(int nz, real q, int nterms) {
-    int nrfft = nz %/% 2 + 1;
-    vector[nrfft] result = zeros_vector(nrfft);
-    vector[nrfft] k = linspaced_vector(nrfft, 0, nrfft - 1);
-    for (n in 0:nterms) {
-        result += q ^ ((k + n * nz) ^ 2) + q ^ ((nz - k + n * nz) ^ 2);
-    }
-    return nz * result;
-}

gptools-stan/gptools/stan/profile/fourier_centered.stan

@@ -13,7 +13,7 @@ parameters {
 }
 
 model {
-    vector[n %/% 2 + 1] cov_rfft = gp_periodic_exp_quad_cov_rfft(n, sigma, length_scale, n, 10);
+    vector[n %/% 2 + 1] cov_rfft = gp_periodic_exp_quad_cov_rfft(n, sigma, length_scale, n);
     eta ~ gp_rfft(zeros_vector(n), cov_rfft);
     y[observed_idx] ~ normal(eta[observed_idx], noise_scale);
 }

gptools-stan/gptools/stan/profile/fourier_non_centered.stan

@@ -15,7 +15,7 @@ parameters {
 transformed parameters {
     vector[n] eta;
     {
-        vector[n %/% 2 + 1] cov_rfft = gp_periodic_exp_quad_cov_rfft(n, sigma, length_scale, n, 10);
+        vector[n %/% 2 + 1] cov_rfft = gp_periodic_exp_quad_cov_rfft(n, sigma, length_scale, n);
         eta = gp_transform_inv_rfft(eta_, zeros_vector(n), cov_rfft);
     }
 }

gptools-stan/tests/test_stan_functions.py

@@ -359,26 +359,6 @@ def assert_stan_python_allclose(
         "desired": np.zeros((m, n)),
     })
 
-for n in [5, 8]:
-    z = np.linspace(0, 1, n, endpoint=False)
-    q = np.random.uniform(0.25, 0.75, n)
-    add_configuration({
-        "stan_function": "jtheta",
-        "arg_types": {"n_": "int", "z": "vector[n_]", "q": "vector[n_]", "nterms": "int"},
-        "arg_values": {"n_": n, "z": z, "q": q, "nterms": 10},
-        "result_type": "vector[n_]",
-        "includes": ["gptools_util.stan"],
-        "desired": kernels.jtheta(z, q),
-    })
-    add_configuration({
-        "stan_function": "jtheta_rfft",
-        "arg_types": {"n": "int", "q": "real", "nterms": "int"},
-        "arg_values": {"n": n, "q": q[0], "nterms": 10},
-        "result_type": "vector[n %/% 2 + 1]",
-        "includes": ["gptools_util.stan"],
-        "desired": kernels.jtheta_rfft(n, q[0]),
-    })
-
 for ndim in [1, 2, 3]:
     n = 1 + np.random.poisson(50)
     m = 1 + np.random.poisson(50)
@@ -392,9 +372,9 @@ def assert_stan_python_allclose(
         "stan_function": "gp_periodic_exp_quad_cov",
         "arg_types": {"n_": "int", "m_": "int", "p_": "int", "x": "array [n_] vector[p_]",
                       "y": "array [m_] vector[p_]", "sigma": "real", "length_scale": "vector[p_]",
-                      "period": "vector[p_]", "nterms": "int"},
+                      "period": "vector[p_]"},
         "arg_values": {"n_": n, "m_": m, "p_": ndim, "x": x, "y": y, "sigma": sigma,
-                       "length_scale": length_scale, "period": period, "nterms": 100},
+                       "length_scale": length_scale, "period": period},
         "result_type": "matrix[n_, m_]",
         "includes": ["gptools_util.stan", "gptools_kernels.stan"],
         "desired": kernel.evaluate(x[:, None], y[None]),
@@ -428,10 +408,8 @@ def assert_stan_python_allclose(
     period = np.random.gamma(100, 0.1)
     add_configuration({
         "stan_function": "gp_periodic_exp_quad_cov_rfft",
-        "arg_types": {"m": "int", "sigma": "real", "length_scale": "real", "period": "real",
-                      "nterms": "int"},
-        "arg_values": {"m": n, "sigma": sigma, "length_scale": length_scale, "period": period,
-                       "nterms": 100},
+        "arg_types": {"m": "int", "sigma": "real", "length_scale": "real", "period": "real"},
+        "arg_values": {"m": n, "sigma": sigma, "length_scale": length_scale, "period": period},
         "result_type": "vector[m %/% 2 + 1]",
         "includes": ["gptools_util.stan", "gptools_kernels.stan"],
         "desired": kernels.ExpQuadKernel(sigma, length_scale, period=period).evaluate_rfft([n]),
@@ -453,9 +431,9 @@ def assert_stan_python_allclose(
         add_configuration({
             "stan_function": "gp_periodic_exp_quad_cov_rfft2",
             "arg_types": {"m": "int", "n": "int", "sigma": "real", "length_scale": "vector[2]",
-                          "period": "vector[2]", "nterms": "int"},
+                          "period": "vector[2]"},
             "arg_values": {"m": m, "n": n, "sigma": sigma, "length_scale": length_scale,
-                           "period": period, "nterms": 100},
+                           "period": period},
             "result_type": "matrix[m, n %/% 2 + 1]",
             "includes": ["gptools_util.stan", "gptools_kernels.stan"],
             "desired": kernels.ExpQuadKernel(sigma, length_scale, period=period)

gptools-util/gptools/util/kernels.py

@@ -1,76 +1,14 @@
 import math
-import numbers
 import numpy as np
 import operator
-from typing import Callable, Optional
+from typing import Callable
 from . import ArrayOrTensor, ArrayOrTensorDispatch, coordgrid, OptionalArrayOrTensor
 from .fft import expand_rfft
 
 
 dispatch = ArrayOrTensorDispatch()
 
 
-def _jtheta_num_terms(q: ArrayOrTensor, rtol: float = 1e-9) -> int:
-    return math.ceil(math.log(rtol) / math.log(dispatch.max(q)))
-
-
-def jtheta(z: ArrayOrTensor, q: ArrayOrTensor, nterms: Optional[int] = None,
-           max_batch_size: int = 1e6) -> ArrayOrTensor:
-    r"""
-    Evaluate the Jacobi theta function using a series approximation.
-
-    .. math::
-
-        \vartheta_3\left(q,z\right) = 1 + 2 \sum_{n=1}^\infty q^{n^2} \cos\left(2\pi n z\right)
-
-    Args:
-        z: Argument of the theta function.
-        q: Nome of the theta function with modulus less than one.
-        nterms: Number of terms in the series approximation (defaults to achieve a relative
-            tolerance of :math:`10^{-9}`, 197 terms for `q = 0.9`).
-        max_batch_size: Maximum number of terms per batch.
-    """
-    # TODO: fix for torch.
-    q, z = np.broadcast_arrays(q, z)
-    nterms = nterms or _jtheta_num_terms(q)
-    # If the dimensions of q and z are large and the number of terms is large, we can run into
-    # memory issues here. We batch the evaluation if necessary to overcome this issue. The maximum
-    # number of terms should be no more than 10 ^ 6 elements (about 8MB at 64-bit precision).
-    batch_size = int(max(1, max_batch_size / (q.size * nterms)))
-    series = 0.0
-    for offset in range(0, nterms, batch_size):
-        size = min(batch_size, nterms - offset)
-        n = 1 + offset + dispatch[z].arange(size)
-        series = series \
-            + (q[..., None] ** (n ** 2) * dispatch.cos(2 * math.pi * z[..., None] * n)).sum(axis=-1)
-
-    return 1 + 2 * series
-
-
-def jtheta_rfft(nz: int, q: ArrayOrTensor, nterms: Optional[int] = None) -> ArrayOrTensor:
-    """
-    Evaluate the real fast Fourier transform of the Jacobi theta function evaluated on the unit
-    interval with `nz` grid points.
-
-    The :func:`jtheta` and :func:`jtheta_rfft` functions are related by
-
-    >>> nz = ...
-    >>> q = ...
-    >>> z = np.linspace(0, 1, nz, endpoint=False)
-    >>> np.fft.rfft(jtheta(z, q)) == jtheta_rfft(nz, q)
-
-    Args:
-        nz: Number of grid points.
-        q: Nome of the theta function with modulus less than one.
-        nterms: Number of terms in the series approximation (defaults to achieve a relative
-            tolerance of :math:`10^{-9}`, 197 terms for `q = 0.9`).
-    """
-    nterms = nterms or _jtheta_num_terms(q)
-    k = np.arange(nz // 2 + 1)
-    ns = nz * np.arange(nterms)[:, None]
-    return nz * ((q ** ((k + ns) ** 2)).sum(axis=0) + (q ** ((nz - k + ns) ** 2)).sum(axis=0))
-
-
 def evaluate_residuals(x: ArrayOrTensor, y: OptionalArrayOrTensor = None,
                        period: OptionalArrayOrTensor = None) -> ArrayOrTensor:
     """
@@ -290,45 +228,28 @@ class ExpQuadKernel(Kernel):
         sigma: Scale of the covariance.
         length_scale: Correlation length.
         period: Period for circular boundary conditions.
-        num_terms: Number of terms in the series approximation of the heat equation solution.
     """
-    def __init__(self, sigma: float, length_scale: float, period: OptionalArrayOrTensor = None,
-                 num_terms: Optional[int] = None) -> None:
+    def __init__(self, sigma: float, length_scale: float, period: OptionalArrayOrTensor = None) \
+            -> None:
         super().__init__(period)
         self.sigma = sigma
         self.length_scale = length_scale
-        if self.is_periodic:
-            # Evaluate the effective relaxation time of the heat kernel.
-            self.time = 2 * (math.pi * self.length_scale / self.period) ** 2
-            if num_terms is None:
-                num_terms = _jtheta_num_terms(dispatch.exp(-self.time).max())
-            if not isinstance(num_terms, numbers.Number):
-                num_terms = max(num_terms)
-            self.num_terms = int(num_terms)
-        else:
-            self.time = self.num_terms = None
 
     def evaluate(self, x: ArrayOrTensor, y: OptionalArrayOrTensor = None) -> ArrayOrTensor:
-        if self.is_periodic:
-            # The residuals will have shape `(..., num_dims)`.
-            residuals = evaluate_residuals(x, y, self.period) / self.period
-            value = jtheta(residuals, dispatch.exp(-self.time), self.num_terms) \
-                * (self.time / math.pi) ** 0.5
-            cov = self.sigma ** 2 * value.prod(axis=-1)
-            return cov
-        else:
-            residuals = evaluate_residuals(x, y) / self.length_scale
-            exponent = - dispatch.square(residuals).sum(axis=-1) / 2
-            return self.sigma * self.sigma * dispatch.exp(exponent)
+        residuals = evaluate_residuals(x, y, self.period) / self.length_scale
+        exponent = - dispatch.square(residuals).sum(axis=-1) / 2
+        return self.sigma * self.sigma * dispatch.exp(exponent)
 
     def evaluate_rfft(self, shape: tuple[int]) -> ArrayOrTensor:
         if not self.is_periodic:
-            raise ValueError("kernel must be periodic")
+            raise NotImplementedError
         ndim = len(shape)
-        time = self.time * np.ones(ndim)
+        rescaled_length_scale = self.length_scale * np.ones(ndim) / self.period
         value = None
         for i, size in enumerate(shape):
-            part = jtheta_rfft(size, np.exp(-time[i])) * (time[i] / math.pi) ** 0.5
+            xi = np.arange(size // 2 + 1)
+            part = size * rescaled_length_scale[i] * (2 * math.pi) ** 0.5 \
+                * np.exp(-2 * (math.pi * xi * rescaled_length_scale[i]) ** 2)
             if i != ndim - 1:
                 part = expand_rfft(part, size)
             if value is None:
@@ -359,22 +280,19 @@ def __init__(self, dof: float, sigma: float, length_scale: float,
         self.dof = dof
 
     def evaluate(self, x: ArrayOrTensor, y: OptionalArrayOrTensor = None) -> ArrayOrTensor:
-        if self.is_periodic:
-            raise NotImplementedError
+        residuals = evaluate_residuals(x, y, self.period) / self.length_scale
+        distance = (2 * self.dof * residuals * residuals).sum(axis=-1) ** 0.5
+        if self.dof == 3 / 2:
+            value = 1 + distance
+        elif self.dof == 5 / 2:
+            value = 1 + distance + distance * distance / 3
         else:
-            residuals = evaluate_residuals(x, y) / self.length_scale
-            distance = (2 * self.dof * residuals * residuals).sum(axis=-1) ** 0.5
-            if self.dof == 3 / 2:
-                value = 1 + distance
-            elif self.dof == 5 / 2:
-                value = 1 + distance + distance * distance / 3
-            else:
-                raise NotImplementedError
-            return self.sigma * self.sigma * value * dispatch.exp(-distance)
+            raise NotImplementedError
+        return self.sigma * self.sigma * value * dispatch.exp(-distance)
 
     def evaluate_rfft(self, shape: tuple[int]):
         if not self.is_periodic:
-            raise ValueError("kernel must be periodic")
+            raise NotImplementedError
         from scipy import special
 
         # Construct the grid to evaluate on.

gptools-util/setup.py

@@ -18,7 +18,6 @@
             "flake8",
             "jupyter",
             "matplotlib",
-            "mpmath",
             "networkx",
             "pytest",
             "pytest-cov",