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# Source code for torch.distributions.wishart

import math
import warnings
from numbers import Number
from typing import Optional, Union

import torch
from torch import nan
from torch.distributions import constraints
from torch.distributions.exp_family import ExponentialFamily
from torch.distributions.multivariate_normal import _precision_to_scale_tril
from torch.distributions.utils import lazy_property

__all__ = ["Wishart"]

_log_2 = math.log(2)

def _mvdigamma(x: torch.Tensor, p: int) -> torch.Tensor:
assert x.gt((p - 1) / 2).all(), "Wrong domain for multivariate digamma function."
x.unsqueeze(-1)
- torch.arange(p, dtype=x.dtype, device=x.device).div(2).expand(x.shape + (-1,))
).sum(-1)

def _clamp_above_eps(x: torch.Tensor) -> torch.Tensor:
# We assume positive input for this function
return x.clamp(min=torch.finfo(x.dtype).eps)

[docs]class Wishart(ExponentialFamily): r""" Creates a Wishart distribution parameterized by a symmetric positive definite matrix :math:\Sigma, or its Cholesky decomposition :math:\mathbf{\Sigma} = \mathbf{L}\mathbf{L}^\top Example: >>> # xdoctest: +SKIP("FIXME: scale_tril must be at least two-dimensional") >>> m = Wishart(torch.Tensor([2]), covariance_matrix=torch.eye(2)) >>> m.sample() # Wishart distributed with mean=df * I and >>> # variance(x_ij)=df for i != j and variance(x_ij)=2 * df for i == j Args: df (float or Tensor): real-valued parameter larger than the (dimension of Square matrix) - 1 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. 'torch.distributions.LKJCholesky' is a restricted Wishart distribution.[1] **References** [1] Wang, Z., Wu, Y. and Chu, H., 2018. On equivalence of the LKJ distribution and the restricted Wishart distribution. [2] Sawyer, S., 2007. Wishart Distributions and Inverse-Wishart Sampling. [3] Anderson, T. W., 2003. An Introduction to Multivariate Statistical Analysis (3rd ed.). [4] Odell, P. L. & Feiveson, A. H., 1966. A Numerical Procedure to Generate a SampleCovariance Matrix. JASA, 61(313):199-203. [5] Ku, Y.-C. & Bloomfield, P., 2010. Generating Random Wishart Matrices with Fractional Degrees of Freedom in OX. """ arg_constraints = { "covariance_matrix": constraints.positive_definite, "precision_matrix": constraints.positive_definite, "scale_tril": constraints.lower_cholesky, "df": constraints.greater_than(0), } support = constraints.positive_definite has_rsample = True _mean_carrier_measure = 0 def __init__( self, df: Union[torch.Tensor, Number], covariance_matrix: Optional[torch.Tensor] = None, precision_matrix: Optional[torch.Tensor] = None, scale_tril: Optional[torch.Tensor] = None, validate_args=None, ): assert (covariance_matrix is not None) + (scale_tril is not None) + ( precision_matrix is not None ) == 1, "Exactly one of covariance_matrix or precision_matrix or scale_tril may be specified." param = next( p for p in (covariance_matrix, precision_matrix, scale_tril) if p is not None ) if param.dim() < 2: raise ValueError( "scale_tril must be at least two-dimensional, with optional leading batch dimensions" ) if isinstance(df, Number): batch_shape = torch.Size(param.shape[:-2]) self.df = torch.tensor(df, dtype=param.dtype, device=param.device) else: batch_shape = torch.broadcast_shapes(param.shape[:-2], df.shape) self.df = df.expand(batch_shape) event_shape = param.shape[-2:] if self.df.le(event_shape[-1] - 1).any(): raise ValueError( f"Value of df={df} expected to be greater than ndim - 1 = {event_shape[-1]-1}." ) if scale_tril is not None: self.scale_tril = param.expand(batch_shape + (-1, -1)) elif covariance_matrix is not None: self.covariance_matrix = param.expand(batch_shape + (-1, -1)) elif precision_matrix is not None: self.precision_matrix = param.expand(batch_shape + (-1, -1)) self.arg_constraints["df"] = constraints.greater_than(event_shape[-1] - 1) if self.df.lt(event_shape[-1]).any(): warnings.warn( "Low df values detected. Singular samples are highly likely to occur for ndim - 1 < df < ndim." ) super().__init__(batch_shape, event_shape, validate_args=validate_args) self._batch_dims = [-(x + 1) for x in range(len(self._batch_shape))] 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) # Chi2 distribution is needed for Bartlett decomposition sampling self._dist_chi2 = torch.distributions.chi2.Chi2( df=( self.df.unsqueeze(-1) - torch.arange( self._event_shape[-1], dtype=self._unbroadcasted_scale_tril.dtype, device=self._unbroadcasted_scale_tril.device, ).expand(batch_shape + (-1,)) ) )
[docs] def expand(self, batch_shape, _instance=None): new = self._get_checked_instance(Wishart, _instance) batch_shape = torch.Size(batch_shape) cov_shape = batch_shape + self.event_shape new._unbroadcasted_scale_tril = self._unbroadcasted_scale_tril.expand(cov_shape) new.df = self.df.expand(batch_shape) new._batch_dims = [-(x + 1) for x in range(len(batch_shape))] 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) # Chi2 distribution is needed for Bartlett decomposition sampling new._dist_chi2 = torch.distributions.chi2.Chi2( df=( new.df.unsqueeze(-1) - torch.arange( self.event_shape[-1], dtype=new._unbroadcasted_scale_tril.dtype, device=new._unbroadcasted_scale_tril.device, ).expand(batch_shape + (-1,)) ) ) super(Wishart, 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 ) @lazy_property def covariance_matrix(self): return ( self._unbroadcasted_scale_tril @ self._unbroadcasted_scale_tril.transpose(-2, -1) ).expand(self._batch_shape + self._event_shape) @lazy_property def precision_matrix(self): identity = torch.eye( self._event_shape[-1], device=self._unbroadcasted_scale_tril.device, dtype=self._unbroadcasted_scale_tril.dtype, ) return torch.cholesky_solve(identity, self._unbroadcasted_scale_tril).expand( self._batch_shape + self._event_shape ) @property def mean(self): return self.df.view(self._batch_shape + (1, 1)) * self.covariance_matrix @property def mode(self): factor = self.df - self.covariance_matrix.shape[-1] - 1 factor[factor <= 0] = nan return factor.view(self._batch_shape + (1, 1)) * self.covariance_matrix @property def variance(self): V = self.covariance_matrix # has shape (batch_shape x event_shape) diag_V = V.diagonal(dim1=-2, dim2=-1) return self.df.view(self._batch_shape + (1, 1)) * ( V.pow(2) + torch.einsum("...i,...j->...ij", diag_V, diag_V) ) def _bartlett_sampling(self, sample_shape=torch.Size()): p = self._event_shape[-1] # has singleton shape # Implemented Sampling using Bartlett decomposition noise = _clamp_above_eps( self._dist_chi2.rsample(sample_shape).sqrt() ).diag_embed(dim1=-2, dim2=-1) i, j = torch.tril_indices(p, p, offset=-1) noise[..., i, j] = torch.randn( torch.Size(sample_shape) + self._batch_shape + (int(p * (p - 1) / 2),), dtype=noise.dtype, device=noise.device, ) chol = self._unbroadcasted_scale_tril @ noise return chol @ chol.transpose(-2, -1)
[docs] def rsample(self, sample_shape=torch.Size(), max_try_correction=None): r""" .. warning:: In some cases, sampling algorithm based on Bartlett decomposition may return singular matrix samples. Several tries to correct singular samples are performed by default, but it may end up returning singular matrix samples. Singular samples may return -inf values in .log_prob(). In those cases, the user should validate the samples and either fix the value of df or adjust max_try_correction value for argument in .rsample accordingly. """ if max_try_correction is None: max_try_correction = 3 if torch._C._get_tracing_state() else 10 sample_shape = torch.Size(sample_shape) sample = self._bartlett_sampling(sample_shape) # Below part is to improve numerical stability temporally and should be removed in the future is_singular = self.support.check(sample) if self._batch_shape: is_singular = is_singular.amax(self._batch_dims) if torch._C._get_tracing_state(): # Less optimized version for JIT for _ in range(max_try_correction): sample_new = self._bartlett_sampling(sample_shape) sample = torch.where(is_singular, sample_new, sample) is_singular = ~self.support.check(sample) if self._batch_shape: is_singular = is_singular.amax(self._batch_dims) else: # More optimized version with data-dependent control flow. if is_singular.any(): warnings.warn("Singular sample detected.") for _ in range(max_try_correction): sample_new = self._bartlett_sampling(is_singular[is_singular].shape) sample[is_singular] = sample_new is_singular_new = ~self.support.check(sample_new) if self._batch_shape: is_singular_new = is_singular_new.amax(self._batch_dims) is_singular[is_singular.clone()] = is_singular_new if not is_singular.any(): break return sample
[docs] def log_prob(self, value): if self._validate_args: self._validate_sample(value) nu = self.df # has shape (batch_shape) p = self._event_shape[-1] # has singleton shape return ( -nu * ( p * _log_2 / 2 + self._unbroadcasted_scale_tril.diagonal(dim1=-2, dim2=-1) .log() .sum(-1) ) - torch.mvlgamma(nu / 2, p=p) + (nu - p - 1) / 2 * torch.linalg.slogdet(value).logabsdet - torch.cholesky_solve(value, self._unbroadcasted_scale_tril) .diagonal(dim1=-2, dim2=-1) .sum(dim=-1) / 2 )
[docs] def entropy(self): nu = self.df # has shape (batch_shape) p = self._event_shape[-1] # has singleton shape V = self.covariance_matrix # has shape (batch_shape x event_shape) return ( (p + 1) * ( p * _log_2 / 2 + self._unbroadcasted_scale_tril.diagonal(dim1=-2, dim2=-1) .log() .sum(-1) ) + torch.mvlgamma(nu / 2, p=p) - (nu - p - 1) / 2 * _mvdigamma(nu / 2, p=p) + nu * p / 2 )
@property def _natural_params(self): nu = self.df # has shape (batch_shape) p = self._event_shape[-1] # has singleton shape return -self.precision_matrix / 2, (nu - p - 1) / 2 def _log_normalizer(self, x, y): p = self._event_shape[-1] return (y + (p + 1) / 2) * ( -torch.linalg.slogdet(-2 * x).logabsdet + _log_2 * p ) + torch.mvlgamma(y + (p + 1) / 2, p=p)

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