Source code for torch.distributions.lowrank_multivariate_normal
# mypy: allow-untyped-defs
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
[docs]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-deterministic")
>>> 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(
f"cov_factor must be a batch of matrices with shape {event_shape[0]} x m"
)
if cov_diag.shape[-1:] != event_shape:
raise ValueError(
f"cov_diag must be a batch of vectors with shape {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(
f"Incompatible batch shapes: loc {loc.shape}, cov_factor {cov_factor.shape}, cov_diag {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().__init__(batch_shape, event_shape, validate_args=validate_args)
[docs] 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
)
[docs] 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
)
[docs] 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)
[docs] 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)
