multivariate_normal.py

import math

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


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 _batch_trtrs_lower
    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.trtrs(flat_x_swap, flat_L, upper=False)[0].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)


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:

        >>> 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
    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.")

        loc_ = loc.unsqueeze(-1)  # temporarily add dim on right
        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")
            self.scale_tril, loc_ = torch.broadcast_tensors(scale_tril, loc_)
        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")
            self.covariance_matrix, loc_ = torch.broadcast_tensors(covariance_matrix, loc_)
        else:
            if precision_matrix.dim() < 2:
                raise ValueError("precision_matrix must be at least two-dimensional, "
                                 "with optional leading batch dimensions")
            self.precision_matrix, loc_ = torch.broadcast_tensors(precision_matrix, loc_)
        self.loc = loc_[..., 0]  # drop rightmost dim

        batch_shape, event_shape = self.loc.shape[:-1], self.loc.shape[-1:]
        super(MultivariateNormal, self).__init__(batch_shape, event_shape, validate_args=validate_args)

        if scale_tril is not None:
            self._unbroadcasted_scale_tril = scale_tril
        else:
            if precision_matrix is not None:
                self.covariance_matrix = torch.inverse(precision_matrix).expand_as(loc_)
            self._unbroadcasted_scale_tril = torch.cholesky(self.covariance_matrix)

    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.transpose(-1, -2))
                .expand(self._batch_shape + self._event_shape + self._event_shape))

    @lazy_property
    def precision_matrix(self):
        # TODO: use `torch.potri` on `scale_tril` once a backwards pass is implemented.
        scale_tril_inv = torch.inverse(self._unbroadcasted_scale_tril)
        return torch.matmul(scale_tril_inv.transpose(-1, -2), scale_tril_inv).expand(
            self._batch_shape + self._event_shape + self._event_shape)

    @property
    def mean(self):
        return self.loc

    @property
    def variance(self):
        return self._unbroadcasted_scale_tril.pow(2).sum(-1).expand(
            self._batch_shape + self._event_shape)

    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)

    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

    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)