forked from google-deepmind/deepmind-research
-
Notifications
You must be signed in to change notification settings - Fork 0
/
train.py
195 lines (167 loc) · 6.33 KB
/
train.py
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
# Copyright 2020 DeepMind Technologies Limited.
#
# Licensed under the Apache License, Version 2.0 (the "License");
# you may not use this file except in compliance with the License.
# You may obtain a copy of the License at
#
# http://www.apache.org/licenses/LICENSE-2.0
#
# Unless required by applicable law or agreed to in writing, software
# distributed under the License is distributed on an "AS IS" BASIS,
# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
# See the License for the specific language governing permissions and
# limitations under the License.
"""Run GEOMANCER on products of synthetic manifolds."""
import re
from absl import app
from absl import flags
from absl import logging
import geomancer
from matplotlib import gridspec
import matplotlib.pyplot as plt
import numpy as np
from scipy.stats import special_ortho_group
from tqdm import tqdm
flags.DEFINE_list('specification', ['S^2', 'S^2'], 'List of submanifolds')
flags.DEFINE_integer('npts', 1000, 'Number of data points')
flags.DEFINE_boolean('rotate', False, 'Apply random rotation to the data')
FLAGS = flags.FLAGS
def make_so_tangent(q):
"""Given an n x n orthonormal matrix, return a basis for its tangent space."""
n = q.shape[0]
assert np.allclose(q.T @ q, np.eye(n), atol=1e-4, rtol=1e-4)
a = np.zeros((n, n))
ii = 0
dq = np.zeros((n, n, n*(n-1)//2))
for i in range(n):
for j in range(i+1, n):
a[i, j] = 1
a[j, i] = -1
dq[..., ii] = a @ q # tangent vectors are skew-symmetric matrix times Q
a[i, j] = 0
a[j, i] = 0
ii += 1
# reshape and orthonormalize the result
return np.linalg.qr(np.reshape(dq, (n**2, n*(n-1)//2)))[0]
def make_sphere_tangent(x):
_, _, v = np.linalg.svd(x[None, :])
return v[:, 1:]
def make_true_tangents(spec, data):
"""Return a set of orthonormal bases, one for each submanifold."""
for i in range(spec.shape[1]):
assert spec[0, i] == 0 or spec[1, i] == 0
so_dim = sum(dim ** 2 for dim in spec[0])
sphere_dim = sum(dim+1 if dim > 0 else 0 for dim in spec[1])
assert so_dim + sphere_dim == data.shape[0]
ii = 0
tangents = []
for i in range(spec.shape[1]):
if spec[0, i] != 0:
dim = spec[0, i]
tangents.append(make_so_tangent(np.reshape(data[ii:ii+dim**2],
(dim, dim))))
ii += dim ** 2
else:
dim = spec[1, i]
tangents.append(make_sphere_tangent(data[ii:ii+dim+1]))
ii += dim + 1
tangents2 = []
for i in range(len(tangents)):
size1 = sum(x.shape[0] for x in tangents[:i])
size2 = sum(x.shape[0] for x in tangents[i+1:])
tangents2.append(np.concatenate(
(np.zeros((size1, tangents[i].shape[1])),
tangents[i],
np.zeros((size2, tangents[i].shape[1]))), axis=0))
return tangents2
def make_product_manifold(specification, npts):
"""Generate data from a product of manifolds with the given specification."""
data = []
tangents = []
latent_dim = 0
spec_array = np.zeros((2, len(specification)), dtype=np.int32)
for i, spec in enumerate(specification):
so_spec = re.search(r'SO\(([0-9]+)\)', spec) # matches "SO(<numbers>)"
sphere_spec = re.search(r'S\^([0-9]+)', spec) # matches "S^<numbers>"
if sphere_spec is not None:
dim = int(sphere_spec.group(1))
spec_array[1, i] = dim
latent_dim += dim
dat = np.random.randn(npts, dim+1)
dat /= np.tile(np.sqrt(np.sum(dat**2, axis=1)[..., None]),
[1, dim+1])
elif so_spec is not None:
dim = int(so_spec.group(1))
spec_array[0, i] = dim
latent_dim += dim * (dim - 1) // 2
dat = [np.ndarray.flatten(special_ortho_group.rvs(dim), order='C')
for _ in range(npts)]
dat = np.stack(dat)
else:
raise ValueError(f'Unrecognized manifold: {spec}')
data.append(dat)
data = np.concatenate(data, axis=1)
for i in range(spec_array.shape[1]):
if spec_array[0, i] != 0:
dim = spec_array[0, i]
tangents.append(np.zeros((npts, data.shape[1], dim * (dim - 1) // 2)))
elif spec_array[1, i] != 0:
dim = spec_array[1, i]
tangents.append(np.zeros((npts, data.shape[1], dim)))
for i in tqdm(range(npts)):
true_tangent = make_true_tangents(spec_array, data[i])
for j in range(len(specification)):
tangents[j][i] = true_tangent[j]
logging.info('Constructed data and true tangents for %s',
' x '.join(specification))
return data, latent_dim, tangents
def main(_):
# Generate data and run GEOMANCER
data, dim, tangents = make_product_manifold(FLAGS.specification, FLAGS.npts)
if FLAGS.rotate:
rot, _ = np.linalg.qr(np.random.randn(data.shape[1], data.shape[1]))
data_rot = data @ rot.T
components, spectrum = geomancer.fit(data_rot, dim)
errors = geomancer.eval_unaligned(data_rot, components, data, tangents)
else:
components, spectrum = geomancer.fit(data, dim)
errors = geomancer.eval_aligned(components, tangents)
# Plot spectrum
plt.figure(figsize=(8, 6))
plt.scatter(np.arange(len(spectrum)), spectrum, s=100)
largest_gap = np.argmax(spectrum[1:]-spectrum[:-1]) + 1
plt.axvline(largest_gap, linewidth=2, c='r')
plt.xticks([])
plt.yticks(fontsize=18)
plt.xlabel('Index', fontsize=24)
plt.ylabel('Eigenvalue', fontsize=24)
plt.title('GeoManCEr Eigenvalue Spectrum', fontsize=24)
# Plot subspace bases
fig = plt.figure(figsize=(8, 6))
bases = components[0]
gs = gridspec.GridSpec(1, len(bases),
width_ratios=[b.shape[1] for b in bases])
for i in range(len(bases)):
ax = plt.subplot(gs[i])
ax.imshow(bases[i])
ax.set_xticks([])
ax.set_yticks([])
ax.set_title(r'$T_{\mathbf{x}_1}\mathcal{M}_%d$' % (i+1), fontsize=18)
fig.canvas.set_window_title('GeoManCEr Results')
# Plot ground truth
fig = plt.figure(figsize=(8, 6))
gs = gridspec.GridSpec(1, len(tangents),
width_ratios=[b.shape[2] for b in tangents])
for i, spec in enumerate(FLAGS.specification):
ax = plt.subplot(gs[i])
ax.imshow(tangents[i][0])
ax.set_xticks([])
ax.set_yticks([])
ax.set_title(r'$T_{\mathbf{x}_1}%s$' % spec, fontsize=18)
fig.canvas.set_window_title('Ground Truth')
logging.info('Error between subspaces: %.2f +/- %.2f radians',
np.mean(errors),
np.std(errors))
plt.show()
if __name__ == '__main__':
app.run(main)