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
196
197
198
199
200
201
202
203
204
205
206
207
208
|
import math
import torch
from torch.distributions import constraints
from torch.distributions.distribution import Distribution
from torch.distributions.multivariate_normal import _batch_mahalanobis, _batch_mv
from torch.distributions.utils import _standard_normal, lazy_property
__all__ = ['LowRankMultivariateNormal']
def _batch_capacitance_tril(W, D):
r"""
Computes Cholesky of :math:`I + W.T @ inv(D) @ W` for a batch of matrices :math:`W`
and a batch of vectors :math:`D`.
"""
m = W.size(-1)
Wt_Dinv = W.mT / D.unsqueeze(-2)
K = torch.matmul(Wt_Dinv, W).contiguous()
K.view(-1, m * m)[:, ::m + 1] += 1 # add identity matrix to K
return torch.linalg.cholesky(K)
def _batch_lowrank_logdet(W, D, capacitance_tril):
r"""
Uses "matrix determinant lemma"::
log|W @ W.T + D| = log|C| + log|D|,
where :math:`C` is the capacitance matrix :math:`I + W.T @ inv(D) @ W`, to compute
the log determinant.
"""
return 2 * capacitance_tril.diagonal(dim1=-2, dim2=-1).log().sum(-1) + D.log().sum(-1)
def _batch_lowrank_mahalanobis(W, D, x, capacitance_tril):
r"""
Uses "Woodbury matrix identity"::
inv(W @ W.T + D) = inv(D) - inv(D) @ W @ inv(C) @ W.T @ inv(D),
where :math:`C` is the capacitance matrix :math:`I + W.T @ inv(D) @ W`, to compute the squared
Mahalanobis distance :math:`x.T @ inv(W @ W.T + D) @ x`.
"""
Wt_Dinv = W.mT / D.unsqueeze(-2)
Wt_Dinv_x = _batch_mv(Wt_Dinv, x)
mahalanobis_term1 = (x.pow(2) / D).sum(-1)
mahalanobis_term2 = _batch_mahalanobis(capacitance_tril, Wt_Dinv_x)
return mahalanobis_term1 - mahalanobis_term2
class LowRankMultivariateNormal(Distribution):
r"""
Creates a multivariate normal distribution with covariance matrix having a low-rank form
parameterized by :attr:`cov_factor` and :attr:`cov_diag`::
covariance_matrix = cov_factor @ cov_factor.T + cov_diag
Example:
>>> # xdoctest: +REQUIRES(env:TORCH_DOCTEST_LAPACK)
>>> # xdoctest: +IGNORE_WANT("non-determenistic")
>>> m = LowRankMultivariateNormal(torch.zeros(2), torch.tensor([[1.], [0.]]), torch.ones(2))
>>> m.sample() # normally distributed with mean=`[0,0]`, cov_factor=`[[1],[0]]`, cov_diag=`[1,1]`
tensor([-0.2102, -0.5429])
Args:
loc (Tensor): mean of the distribution with shape `batch_shape + event_shape`
cov_factor (Tensor): factor part of low-rank form of covariance matrix with shape
`batch_shape + event_shape + (rank,)`
cov_diag (Tensor): diagonal part of low-rank form of covariance matrix with shape
`batch_shape + event_shape`
Note:
The computation for determinant and inverse of covariance matrix is avoided when
`cov_factor.shape[1] << cov_factor.shape[0]` thanks to `Woodbury matrix identity
<https://en.wikipedia.org/wiki/Woodbury_matrix_identity>`_ and
`matrix determinant lemma <https://en.wikipedia.org/wiki/Matrix_determinant_lemma>`_.
Thanks to these formulas, we just need to compute the determinant and inverse of
the small size "capacitance" matrix::
capacitance = I + cov_factor.T @ inv(cov_diag) @ cov_factor
"""
arg_constraints = {"loc": constraints.real_vector,
"cov_factor": constraints.independent(constraints.real, 2),
"cov_diag": constraints.independent(constraints.positive, 1)}
support = constraints.real_vector
has_rsample = True
def __init__(self, loc, cov_factor, cov_diag, validate_args=None):
if loc.dim() < 1:
raise ValueError("loc must be at least one-dimensional.")
event_shape = loc.shape[-1:]
if cov_factor.dim() < 2:
raise ValueError("cov_factor must be at least two-dimensional, "
"with optional leading batch dimensions")
if cov_factor.shape[-2:-1] != event_shape:
raise ValueError("cov_factor must be a batch of matrices with shape {} x m"
.format(event_shape[0]))
if cov_diag.shape[-1:] != event_shape:
raise ValueError("cov_diag must be a batch of vectors with shape {}".format(event_shape))
loc_ = loc.unsqueeze(-1)
cov_diag_ = cov_diag.unsqueeze(-1)
try:
loc_, self.cov_factor, cov_diag_ = torch.broadcast_tensors(loc_, cov_factor, cov_diag_)
except RuntimeError as e:
raise ValueError("Incompatible batch shapes: loc {}, cov_factor {}, cov_diag {}"
.format(loc.shape, cov_factor.shape, cov_diag.shape)) from e
self.loc = loc_[..., 0]
self.cov_diag = cov_diag_[..., 0]
batch_shape = self.loc.shape[:-1]
self._unbroadcasted_cov_factor = cov_factor
self._unbroadcasted_cov_diag = cov_diag
self._capacitance_tril = _batch_capacitance_tril(cov_factor, cov_diag)
super(LowRankMultivariateNormal, self).__init__(batch_shape, event_shape,
validate_args=validate_args)
def expand(self, batch_shape, _instance=None):
new = self._get_checked_instance(LowRankMultivariateNormal, _instance)
batch_shape = torch.Size(batch_shape)
loc_shape = batch_shape + self.event_shape
new.loc = self.loc.expand(loc_shape)
new.cov_diag = self.cov_diag.expand(loc_shape)
new.cov_factor = self.cov_factor.expand(loc_shape + self.cov_factor.shape[-1:])
new._unbroadcasted_cov_factor = self._unbroadcasted_cov_factor
new._unbroadcasted_cov_diag = self._unbroadcasted_cov_diag
new._capacitance_tril = self._capacitance_tril
super(LowRankMultivariateNormal, new).__init__(batch_shape,
self.event_shape,
validate_args=False)
new._validate_args = self._validate_args
return new
@property
def mean(self):
return self.loc
@property
def mode(self):
return self.loc
@lazy_property
def variance(self):
return (self._unbroadcasted_cov_factor.pow(2).sum(-1)
+ self._unbroadcasted_cov_diag).expand(self._batch_shape + self._event_shape)
@lazy_property
def scale_tril(self):
# The following identity is used to increase the numerically computation stability
# for Cholesky decomposition (see http://www.gaussianprocess.org/gpml/, Section 3.4.3):
# W @ W.T + D = D1/2 @ (I + D-1/2 @ W @ W.T @ D-1/2) @ D1/2
# The matrix "I + D-1/2 @ W @ W.T @ D-1/2" has eigenvalues bounded from below by 1,
# hence it is well-conditioned and safe to take Cholesky decomposition.
n = self._event_shape[0]
cov_diag_sqrt_unsqueeze = self._unbroadcasted_cov_diag.sqrt().unsqueeze(-1)
Dinvsqrt_W = self._unbroadcasted_cov_factor / cov_diag_sqrt_unsqueeze
K = torch.matmul(Dinvsqrt_W, Dinvsqrt_W.mT).contiguous()
K.view(-1, n * n)[:, ::n + 1] += 1 # add identity matrix to K
scale_tril = cov_diag_sqrt_unsqueeze * torch.linalg.cholesky(K)
return scale_tril.expand(self._batch_shape + self._event_shape + self._event_shape)
@lazy_property
def covariance_matrix(self):
covariance_matrix = (torch.matmul(self._unbroadcasted_cov_factor,
self._unbroadcasted_cov_factor.mT)
+ torch.diag_embed(self._unbroadcasted_cov_diag))
return covariance_matrix.expand(self._batch_shape + self._event_shape +
self._event_shape)
@lazy_property
def precision_matrix(self):
# We use "Woodbury matrix identity" to take advantage of low rank form::
# inv(W @ W.T + D) = inv(D) - inv(D) @ W @ inv(C) @ W.T @ inv(D)
# where :math:`C` is the capacitance matrix.
Wt_Dinv = (self._unbroadcasted_cov_factor.mT
/ self._unbroadcasted_cov_diag.unsqueeze(-2))
A = torch.linalg.solve_triangular(self._capacitance_tril, Wt_Dinv, upper=False)
precision_matrix = torch.diag_embed(self._unbroadcasted_cov_diag.reciprocal()) - A.mT @ A
return precision_matrix.expand(self._batch_shape + self._event_shape +
self._event_shape)
def rsample(self, sample_shape=torch.Size()):
shape = self._extended_shape(sample_shape)
W_shape = shape[:-1] + self.cov_factor.shape[-1:]
eps_W = _standard_normal(W_shape, dtype=self.loc.dtype, device=self.loc.device)
eps_D = _standard_normal(shape, dtype=self.loc.dtype, device=self.loc.device)
return (self.loc + _batch_mv(self._unbroadcasted_cov_factor, eps_W)
+ self._unbroadcasted_cov_diag.sqrt() * eps_D)
def log_prob(self, value):
if self._validate_args:
self._validate_sample(value)
diff = value - self.loc
M = _batch_lowrank_mahalanobis(self._unbroadcasted_cov_factor,
self._unbroadcasted_cov_diag,
diff,
self._capacitance_tril)
log_det = _batch_lowrank_logdet(self._unbroadcasted_cov_factor,
self._unbroadcasted_cov_diag,
self._capacitance_tril)
return -0.5 * (self._event_shape[0] * math.log(2 * math.pi) + log_det + M)
def entropy(self):
log_det = _batch_lowrank_logdet(self._unbroadcasted_cov_factor,
self._unbroadcasted_cov_diag,
self._capacitance_tril)
H = 0.5 * (self._event_shape[0] * (1.0 + math.log(2 * math.pi)) + log_det)
if len(self._batch_shape) == 0:
return H
else:
return H.expand(self._batch_shape)
|
