143 lines
6.2 KiB
Python
143 lines
6.2 KiB
Python
"""
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This closely follows the implementation in NumPyro (https://github.com/pyro-ppl/numpyro).
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Original copyright notice:
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# Copyright: Contributors to the Pyro project.
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# SPDX-License-Identifier: Apache-2.0
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"""
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import math
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import torch
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from torch.distributions import Beta, constraints
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from torch.distributions.distribution import Distribution
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from torch.distributions.utils import broadcast_all
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__all__ = ["LKJCholesky"]
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class LKJCholesky(Distribution):
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r"""
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LKJ distribution for lower Cholesky factor of correlation matrices.
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The distribution is controlled by ``concentration`` parameter :math:`\eta`
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to make the probability of the correlation matrix :math:`M` generated from
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a Cholesky factor proportional to :math:`\det(M)^{\eta - 1}`. Because of that,
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when ``concentration == 1``, we have a uniform distribution over Cholesky
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factors of correlation matrices::
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L ~ LKJCholesky(dim, concentration)
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X = L @ L' ~ LKJCorr(dim, concentration)
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Note that this distribution samples the
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Cholesky factor of correlation matrices and not the correlation matrices
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themselves and thereby differs slightly from the derivations in [1] for
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the `LKJCorr` distribution. For sampling, this uses the Onion method from
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[1] Section 3.
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Example::
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>>> # xdoctest: +IGNORE_WANT("non-deterministic")
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>>> l = LKJCholesky(3, 0.5)
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>>> l.sample() # l @ l.T is a sample of a correlation 3x3 matrix
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tensor([[ 1.0000, 0.0000, 0.0000],
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[ 0.3516, 0.9361, 0.0000],
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[-0.1899, 0.4748, 0.8593]])
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Args:
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dimension (dim): dimension of the matrices
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concentration (float or Tensor): concentration/shape parameter of the
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distribution (often referred to as eta)
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**References**
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[1] `Generating random correlation matrices based on vines and extended onion method` (2009),
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Daniel Lewandowski, Dorota Kurowicka, Harry Joe.
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Journal of Multivariate Analysis. 100. 10.1016/j.jmva.2009.04.008
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"""
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arg_constraints = {"concentration": constraints.positive}
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support = constraints.corr_cholesky
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def __init__(self, dim, concentration=1.0, validate_args=None):
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if dim < 2:
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raise ValueError(
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f"Expected dim to be an integer greater than or equal to 2. Found dim={dim}."
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)
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self.dim = dim
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(self.concentration,) = broadcast_all(concentration)
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batch_shape = self.concentration.size()
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event_shape = torch.Size((dim, dim))
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# This is used to draw vectorized samples from the beta distribution in Sec. 3.2 of [1].
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marginal_conc = self.concentration + 0.5 * (self.dim - 2)
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offset = torch.arange(
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self.dim - 1,
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dtype=self.concentration.dtype,
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device=self.concentration.device,
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)
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offset = torch.cat([offset.new_zeros((1,)), offset])
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beta_conc1 = offset + 0.5
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beta_conc0 = marginal_conc.unsqueeze(-1) - 0.5 * offset
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self._beta = Beta(beta_conc1, beta_conc0)
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super().__init__(batch_shape, event_shape, validate_args)
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def expand(self, batch_shape, _instance=None):
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new = self._get_checked_instance(LKJCholesky, _instance)
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batch_shape = torch.Size(batch_shape)
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new.dim = self.dim
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new.concentration = self.concentration.expand(batch_shape)
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new._beta = self._beta.expand(batch_shape + (self.dim,))
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super(LKJCholesky, new).__init__(
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batch_shape, self.event_shape, validate_args=False
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)
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new._validate_args = self._validate_args
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return new
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def sample(self, sample_shape=torch.Size()):
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# This uses the Onion method, but there are a few differences from [1] Sec. 3.2:
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# - This vectorizes the for loop and also works for heterogeneous eta.
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# - Same algorithm generalizes to n=1.
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# - The procedure is simplified since we are sampling the cholesky factor of
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# the correlation matrix instead of the correlation matrix itself. As such,
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# we only need to generate `w`.
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y = self._beta.sample(sample_shape).unsqueeze(-1)
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u_normal = torch.randn(
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self._extended_shape(sample_shape), dtype=y.dtype, device=y.device
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).tril(-1)
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u_hypersphere = u_normal / u_normal.norm(dim=-1, keepdim=True)
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# Replace NaNs in first row
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u_hypersphere[..., 0, :].fill_(0.0)
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w = torch.sqrt(y) * u_hypersphere
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# Fill diagonal elements; clamp for numerical stability
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eps = torch.finfo(w.dtype).tiny
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diag_elems = torch.clamp(1 - torch.sum(w**2, dim=-1), min=eps).sqrt()
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w += torch.diag_embed(diag_elems)
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return w
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def log_prob(self, value):
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# See: https://mc-stan.org/docs/2_25/functions-reference/cholesky-lkj-correlation-distribution.html
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# The probability of a correlation matrix is proportional to
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# determinant ** (concentration - 1) = prod(L_ii ^ 2(concentration - 1))
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# Additionally, the Jacobian of the transformation from Cholesky factor to
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# correlation matrix is:
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# prod(L_ii ^ (D - i))
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# So the probability of a Cholesky factor is propotional to
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# prod(L_ii ^ (2 * concentration - 2 + D - i)) = prod(L_ii ^ order_i)
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# with order_i = 2 * concentration - 2 + D - i
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if self._validate_args:
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self._validate_sample(value)
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diag_elems = value.diagonal(dim1=-1, dim2=-2)[..., 1:]
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order = torch.arange(2, self.dim + 1, device=self.concentration.device)
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order = 2 * (self.concentration - 1).unsqueeze(-1) + self.dim - order
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unnormalized_log_pdf = torch.sum(order * diag_elems.log(), dim=-1)
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# Compute normalization constant (page 1999 of [1])
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dm1 = self.dim - 1
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alpha = self.concentration + 0.5 * dm1
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denominator = torch.lgamma(alpha) * dm1
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numerator = torch.mvlgamma(alpha - 0.5, dm1)
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# pi_constant in [1] is D * (D - 1) / 4 * log(pi)
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# pi_constant in multigammaln is (D - 1) * (D - 2) / 4 * log(pi)
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# hence, we need to add a pi_constant = (D - 1) * log(pi) / 2
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pi_constant = 0.5 * dm1 * math.log(math.pi)
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normalize_term = pi_constant + numerator - denominator
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return unnormalized_log_pdf - normalize_term
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