Source code for torch.distributions.multivariate_normal

import math

import torch
from torch.distributions import constraints
from torch.distributions.distribution import Distribution
from torch.distributions.utils import _standard_normal, lazy_property

__all__ = ['MultivariateNormal']

def _batch_mv(bmat, bvec):
    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):
    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)) +
    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:
    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-determenistic") >>> 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=torch.Size()): 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)


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