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| """ | |
| 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"] | |
| 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) | |
| 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 | |
| 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 | |
| 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 | |