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Source code for torch.distributions.lkj_cholesky

"""
This closely follows the implementation in NumPyro (https://github.com/pyro-ppl/numpyro).

Original copyright notice:

# Copyright: Contributors to the Pyro project.
# SPDX-License-Identifier: Apache-2.0
"""

import math

import torch
from torch.distributions import Beta, constraints
from torch.distributions.distribution import Distribution
from torch.distributions.utils import broadcast_all

__all__ = ["LKJCholesky"]


[docs]class LKJCholesky(Distribution): r""" LKJ distribution for lower Cholesky factor of correlation matrices. The distribution is controlled by ``concentration`` parameter :math:`\eta` to make the probability of the correlation matrix :math:`M` generated from a Cholesky factor proportional to :math:`\det(M)^{\eta - 1}`. Because of that, when ``concentration == 1``, we have a uniform distribution over Cholesky factors of correlation matrices:: L ~ LKJCholesky(dim, concentration) X = L @ L' ~ LKJCorr(dim, concentration) Note that this distribution samples the Cholesky factor of correlation matrices and not the correlation matrices themselves and thereby differs slightly from the derivations in [1] for the `LKJCorr` distribution. For sampling, this uses the Onion method from [1] Section 3. Example:: >>> # xdoctest: +IGNORE_WANT("non-deterministic") >>> l = LKJCholesky(3, 0.5) >>> l.sample() # l @ l.T is a sample of a correlation 3x3 matrix tensor([[ 1.0000, 0.0000, 0.0000], [ 0.3516, 0.9361, 0.0000], [-0.1899, 0.4748, 0.8593]]) Args: dimension (dim): dimension of the matrices concentration (float or Tensor): concentration/shape parameter of the distribution (often referred to as eta) **References** [1] `Generating random correlation matrices based on vines and extended onion method` (2009), Daniel Lewandowski, Dorota Kurowicka, Harry Joe. Journal of Multivariate Analysis. 100. 10.1016/j.jmva.2009.04.008 """ arg_constraints = {"concentration": constraints.positive} support = constraints.corr_cholesky def __init__(self, dim, concentration=1.0, validate_args=None): if dim < 2: raise ValueError( f"Expected dim to be an integer greater than or equal to 2. Found dim={dim}." ) self.dim = dim (self.concentration,) = broadcast_all(concentration) batch_shape = self.concentration.size() event_shape = torch.Size((dim, dim)) # This is used to draw vectorized samples from the beta distribution in Sec. 3.2 of [1]. marginal_conc = self.concentration + 0.5 * (self.dim - 2) offset = torch.arange( self.dim - 1, dtype=self.concentration.dtype, device=self.concentration.device, ) offset = torch.cat([offset.new_zeros((1,)), offset]) beta_conc1 = offset + 0.5 beta_conc0 = marginal_conc.unsqueeze(-1) - 0.5 * offset self._beta = Beta(beta_conc1, beta_conc0) super().__init__(batch_shape, event_shape, validate_args)
[docs] def expand(self, batch_shape, _instance=None): new = self._get_checked_instance(LKJCholesky, _instance) batch_shape = torch.Size(batch_shape) new.dim = self.dim new.concentration = self.concentration.expand(batch_shape) new._beta = self._beta.expand(batch_shape + (self.dim,)) super(LKJCholesky, new).__init__( batch_shape, self.event_shape, validate_args=False ) new._validate_args = self._validate_args return new
[docs] def sample(self, sample_shape=torch.Size()): # This uses the Onion method, but there are a few differences from [1] Sec. 3.2: # - This vectorizes the for loop and also works for heterogeneous eta. # - Same algorithm generalizes to n=1. # - The procedure is simplified since we are sampling the cholesky factor of # the correlation matrix instead of the correlation matrix itself. As such, # we only need to generate `w`. y = self._beta.sample(sample_shape).unsqueeze(-1) u_normal = torch.randn( self._extended_shape(sample_shape), dtype=y.dtype, device=y.device ).tril(-1) u_hypersphere = u_normal / u_normal.norm(dim=-1, keepdim=True) # Replace NaNs in first row u_hypersphere[..., 0, :].fill_(0.0) w = torch.sqrt(y) * u_hypersphere # Fill diagonal elements; clamp for numerical stability eps = torch.finfo(w.dtype).tiny diag_elems = torch.clamp(1 - torch.sum(w**2, dim=-1), min=eps).sqrt() w += torch.diag_embed(diag_elems) return w
[docs] def log_prob(self, value): # See: https://mc-stan.org/docs/2_25/functions-reference/cholesky-lkj-correlation-distribution.html # The probability of a correlation matrix is proportional to # determinant ** (concentration - 1) = prod(L_ii ^ 2(concentration - 1)) # Additionally, the Jacobian of the transformation from Cholesky factor to # correlation matrix is: # prod(L_ii ^ (D - i)) # So the probability of a Cholesky factor is propotional to # prod(L_ii ^ (2 * concentration - 2 + D - i)) = prod(L_ii ^ order_i) # with order_i = 2 * concentration - 2 + D - i if self._validate_args: self._validate_sample(value) diag_elems = value.diagonal(dim1=-1, dim2=-2)[..., 1:] order = torch.arange(2, self.dim + 1, device=self.concentration.device) order = 2 * (self.concentration - 1).unsqueeze(-1) + self.dim - order unnormalized_log_pdf = torch.sum(order * diag_elems.log(), dim=-1) # Compute normalization constant (page 1999 of [1]) dm1 = self.dim - 1 alpha = self.concentration + 0.5 * dm1 denominator = torch.lgamma(alpha) * dm1 numerator = torch.mvlgamma(alpha - 0.5, dm1) # pi_constant in [1] is D * (D - 1) / 4 * log(pi) # pi_constant in multigammaln is (D - 1) * (D - 2) / 4 * log(pi) # hence, we need to add a pi_constant = (D - 1) * log(pi) / 2 pi_constant = 0.5 * dm1 * math.log(math.pi) normalize_term = pi_constant + numerator - denominator return unnormalized_log_pdf - normalize_term

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