Source code for torch.distributions.multivariate_normal
# mypy: allow-untyped-defs
import math
import torch
from torch.distributions import constraints
from torch.distributions.distribution import Distribution
from torch.distributions.utils import _standard_normal, lazy_property
from torch.types import _size
__all__ = ["MultivariateNormal"]
def _batch_mv(bmat, bvec):
r"""
Performs a batched matrix-vector product, with compatible but different batch shapes.
This function takes as input `bmat`, containing :math:`n \times n` matrices, and
`bvec`, containing length :math:`n` vectors.
Both `bmat` and `bvec` may have any number of leading dimensions, which correspond
to a batch shape. They are not necessarily assumed to have the same batch shape,
just ones which can be broadcasted.
"""
return torch.matmul(bmat, bvec.unsqueeze(-1)).squeeze(-1)
def _batch_mahalanobis(bL, bx):
r"""
Computes the squared Mahalanobis distance :math:`\mathbf{x}^\top\mathbf{M}^{-1}\mathbf{x}`
for a factored :math:`\mathbf{M} = \mathbf{L}\mathbf{L}^\top`.
Accepts batches for both bL and bx. They are not necessarily assumed to have the same batch
shape, but `bL` one should be able to broadcasted to `bx` one.
"""
n = bx.size(-1)
bx_batch_shape = bx.shape[:-1]
# Assume that bL.shape = (i, 1, n, n), bx.shape = (..., i, j, n),
# we are going to make bx have shape (..., 1, j, i, 1, n) to apply batched tri.solve
bx_batch_dims = len(bx_batch_shape)
bL_batch_dims = bL.dim() - 2
outer_batch_dims = bx_batch_dims - bL_batch_dims
old_batch_dims = outer_batch_dims + bL_batch_dims
new_batch_dims = outer_batch_dims + 2 * bL_batch_dims
# Reshape bx with the shape (..., 1, i, j, 1, n)
bx_new_shape = bx.shape[:outer_batch_dims]
for sL, sx in zip(bL.shape[:-2], bx.shape[outer_batch_dims:-1]):
bx_new_shape += (sx // sL, sL)
bx_new_shape += (n,)
bx = bx.reshape(bx_new_shape)
# Permute bx to make it have shape (..., 1, j, i, 1, n)
permute_dims = (
list(range(outer_batch_dims))
+ list(range(outer_batch_dims, new_batch_dims, 2))
+ list(range(outer_batch_dims + 1, new_batch_dims, 2))
+ [new_batch_dims]
)
bx = bx.permute(permute_dims)
flat_L = bL.reshape(-1, n, n) # shape = b x n x n
flat_x = bx.reshape(-1, flat_L.size(0), n) # shape = c x b x n
flat_x_swap = flat_x.permute(1, 2, 0) # shape = b x n x c
M_swap = (
torch.linalg.solve_triangular(flat_L, flat_x_swap, upper=False).pow(2).sum(-2)
) # shape = b x c
M = M_swap.t() # shape = c x b
# Now we revert the above reshape and permute operators.
permuted_M = M.reshape(bx.shape[:-1]) # shape = (..., 1, j, i, 1)
permute_inv_dims = list(range(outer_batch_dims))
for i in range(bL_batch_dims):
permute_inv_dims += [outer_batch_dims + i, old_batch_dims + i]
reshaped_M = permuted_M.permute(permute_inv_dims) # shape = (..., 1, i, j, 1)
return reshaped_M.reshape(bx_batch_shape)
def _precision_to_scale_tril(P):
# Ref: https://nbviewer.jupyter.org/gist/fehiepsi/5ef8e09e61604f10607380467eb82006#Precision-to-scale_tril
Lf = torch.linalg.cholesky(torch.flip(P, (-2, -1)))
L_inv = torch.transpose(torch.flip(Lf, (-2, -1)), -2, -1)
Id = torch.eye(P.shape[-1], dtype=P.dtype, device=P.device)
L = torch.linalg.solve_triangular(L_inv, Id, upper=False)
return L
[docs]class MultivariateNormal(Distribution):
r"""
Creates a multivariate normal (also called Gaussian) distribution
parameterized by a mean vector and a covariance matrix.
The multivariate normal distribution can be parameterized either
in terms of a positive definite covariance matrix :math:`\mathbf{\Sigma}`
or a positive definite precision matrix :math:`\mathbf{\Sigma}^{-1}`
or a lower-triangular matrix :math:`\mathbf{L}` with positive-valued
diagonal entries, such that
:math:`\mathbf{\Sigma} = \mathbf{L}\mathbf{L}^\top`. This triangular matrix
can be obtained via e.g. Cholesky decomposition of the covariance.
Example:
>>> # xdoctest: +REQUIRES(env:TORCH_DOCTEST_LAPACK)
>>> # xdoctest: +IGNORE_WANT("non-deterministic")
>>> m = MultivariateNormal(torch.zeros(2), torch.eye(2))
>>> m.sample() # normally distributed with mean=`[0,0]` and covariance_matrix=`I`
tensor([-0.2102, -0.5429])
Args:
loc (Tensor): mean of the distribution
covariance_matrix (Tensor): positive-definite covariance matrix
precision_matrix (Tensor): positive-definite precision matrix
scale_tril (Tensor): lower-triangular factor of covariance, with positive-valued diagonal
Note:
Only one of :attr:`covariance_matrix` or :attr:`precision_matrix` or
:attr:`scale_tril` can be specified.
Using :attr:`scale_tril` will be more efficient: all computations internally
are based on :attr:`scale_tril`. If :attr:`covariance_matrix` or
:attr:`precision_matrix` is passed instead, it is only used to compute
the corresponding lower triangular matrices using a Cholesky decomposition.
"""
arg_constraints = {
"loc": constraints.real_vector,
"covariance_matrix": constraints.positive_definite,
"precision_matrix": constraints.positive_definite,
"scale_tril": constraints.lower_cholesky,
}
support = constraints.real_vector
has_rsample = True
def __init__(
self,
loc,
covariance_matrix=None,
precision_matrix=None,
scale_tril=None,
validate_args=None,
):
if loc.dim() < 1:
raise ValueError("loc must be at least one-dimensional.")
if (covariance_matrix is not None) + (scale_tril is not None) + (
precision_matrix is not None
) != 1:
raise ValueError(
"Exactly one of covariance_matrix or precision_matrix or scale_tril may be specified."
)
if scale_tril is not None:
if scale_tril.dim() < 2:
raise ValueError(
"scale_tril matrix must be at least two-dimensional, "
"with optional leading batch dimensions"
)
batch_shape = torch.broadcast_shapes(scale_tril.shape[:-2], loc.shape[:-1])
self.scale_tril = scale_tril.expand(batch_shape + (-1, -1))
elif covariance_matrix is not None:
if covariance_matrix.dim() < 2:
raise ValueError(
"covariance_matrix must be at least two-dimensional, "
"with optional leading batch dimensions"
)
batch_shape = torch.broadcast_shapes(
covariance_matrix.shape[:-2], loc.shape[:-1]
)
self.covariance_matrix = covariance_matrix.expand(batch_shape + (-1, -1))
else:
if precision_matrix.dim() < 2:
raise ValueError(
"precision_matrix must be at least two-dimensional, "
"with optional leading batch dimensions"
)
batch_shape = torch.broadcast_shapes(
precision_matrix.shape[:-2], loc.shape[:-1]
)
self.precision_matrix = precision_matrix.expand(batch_shape + (-1, -1))
self.loc = loc.expand(batch_shape + (-1,))
event_shape = self.loc.shape[-1:]
super().__init__(batch_shape, event_shape, validate_args=validate_args)
if scale_tril is not None:
self._unbroadcasted_scale_tril = scale_tril
elif covariance_matrix is not None:
self._unbroadcasted_scale_tril = torch.linalg.cholesky(covariance_matrix)
else: # precision_matrix is not None
self._unbroadcasted_scale_tril = _precision_to_scale_tril(precision_matrix)
[docs] def expand(self, batch_shape, _instance=None):
new = self._get_checked_instance(MultivariateNormal, _instance)
batch_shape = torch.Size(batch_shape)
loc_shape = batch_shape + self.event_shape
cov_shape = batch_shape + self.event_shape + self.event_shape
new.loc = self.loc.expand(loc_shape)
new._unbroadcasted_scale_tril = self._unbroadcasted_scale_tril
if "covariance_matrix" in self.__dict__:
new.covariance_matrix = self.covariance_matrix.expand(cov_shape)
if "scale_tril" in self.__dict__:
new.scale_tril = self.scale_tril.expand(cov_shape)
if "precision_matrix" in self.__dict__:
new.precision_matrix = self.precision_matrix.expand(cov_shape)
super(MultivariateNormal, new).__init__(
batch_shape, self.event_shape, validate_args=False
)
new._validate_args = self._validate_args
return new
@lazy_property
def scale_tril(self):
return self._unbroadcasted_scale_tril.expand(
self._batch_shape + self._event_shape + self._event_shape
)
@lazy_property
def covariance_matrix(self):
return torch.matmul(
self._unbroadcasted_scale_tril, self._unbroadcasted_scale_tril.mT
).expand(self._batch_shape + self._event_shape + self._event_shape)
@lazy_property
def precision_matrix(self):
return torch.cholesky_inverse(self._unbroadcasted_scale_tril).expand(
self._batch_shape + self._event_shape + self._event_shape
)
@property
def mean(self):
return self.loc
@property
def mode(self):
return self.loc
@property
def variance(self):
return (
self._unbroadcasted_scale_tril.pow(2)
.sum(-1)
.expand(self._batch_shape + self._event_shape)
)
[docs] def rsample(self, sample_shape: _size = torch.Size()) -> torch.Tensor:
shape = self._extended_shape(sample_shape)
eps = _standard_normal(shape, dtype=self.loc.dtype, device=self.loc.device)
return self.loc + _batch_mv(self._unbroadcasted_scale_tril, eps)
[docs] def log_prob(self, value):
if self._validate_args:
self._validate_sample(value)
diff = value - self.loc
M = _batch_mahalanobis(self._unbroadcasted_scale_tril, diff)
half_log_det = (
self._unbroadcasted_scale_tril.diagonal(dim1=-2, dim2=-1).log().sum(-1)
)
return -0.5 * (self._event_shape[0] * math.log(2 * math.pi) + M) - half_log_det
[docs] def entropy(self):
half_log_det = (
self._unbroadcasted_scale_tril.diagonal(dim1=-2, dim2=-1).log().sum(-1)
)
H = 0.5 * self._event_shape[0] * (1.0 + math.log(2 * math.pi)) + half_log_det
if len(self._batch_shape) == 0:
return H
else:
return H.expand(self._batch_shape)
