lowrank_multivariate_normal.py

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


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.transpose(-1, -2) / 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.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.transpose(-1, -2) / 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:

        >>> m = LowRankMultivariateNormal(torch.zeros(2), torch.tensor([1, 0]), torch.tensor([1, 1]))
        >>> 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,
                       "cov_factor": constraints.real,
                       "cov_diag": constraints.positive}
    support = constraints.real
    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:
            raise ValueError("Incompatible batch shapes: loc {}, cov_factor {}, cov_diag {}"
                             .format(loc.shape, cov_factor.shape, cov_diag.shape))
        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

    @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.transpose(-1, -2)).contiguous()
        K.view(-1, n * n)[:, ::n + 1] += 1  # add identity matrix to K
        scale_tril = cov_diag_sqrt_unsqueeze * torch.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.transpose(-1, -2))
                             + 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.transpose(-1, -2)
                   / self._unbroadcasted_cov_diag.unsqueeze(-2))
        A = torch.trtrs(Wt_Dinv, self._capacitance_tril, upper=False)[0]
        precision_matrix = (torch.diag_embed(self._unbroadcasted_cov_diag.reciprocal())
                            - torch.matmul(A.transpose(-1, -2), 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)