# Source code for pyro.distributions.transforms.polynomial

import math

import torch
import torch.nn as nn
from torch.distributions import constraints

from pyro.distributions.torch_transform import TransformModule
from pyro.distributions.util import copy_docs_from

[docs]@copy_docs_from(TransformModule) class PolynomialFlow(TransformModule): """ An autoregressive normalizing flow as described in Jaini et al. (2019) using the element-wise transformation :math:y_n = c_n + \\int^{x_n}_0\\sum^K_{k=1}\\left(\\sum^R_{r=0}a^{(n)}_{r,k}u^r\\right)du where :math:x_n is the :math:nth input, :math:y_n is the :math:nth output, and :math:c_n\\in\\mathbb{R}, :math:\\left{a^{(n)}_{r,k}\\in\\mathbb{R}\\right} are learnable parameters that are the output of an autoregressive NN inputting :math:x_{\\prec n}={x_1,x_2,\\ldots,x_{n-1}}. Together with TransformedDistribution this provides a way to create richer variational approximations. Example usage: >>> from pyro.nn import AutoRegressiveNN >>> input_dim = 10 >>> count_degree = 4 >>> count_sum = 3 >>> base_dist = dist.Normal(torch.zeros(input_dim), torch.ones(input_dim)) >>> arn = AutoRegressiveNN(input_dim, [input_dim*10], param_dims=[(count_degree + 1)*count_sum]) >>> flow = PolynomialFlow(arn, input_dim=input_dim, count_degree=count_degree, count_sum=count_sum) >>> pyro.module("my_flow", flow) # doctest: +SKIP >>> flow_dist = dist.TransformedDistribution(base_dist, [flow]) >>> flow_dist.sample() # doctest: +SKIP tensor([-0.4071, -0.5030, 0.7924, -0.2366, -0.2387, -0.1417, 0.0868, 0.1389, -0.4629, 0.0986]) The inverse of this transform does not possess an analytical solution and is left unimplemented. However, the inverse is cached when the forward operation is called during sampling, and so samples drawn using polynomial flow can be scored. :param autoregressive_nn: an autoregressive neural network whose forward call returns a tensor of real-valued numbers of size (batch_size, (count_degree+1)*count_sum, input_dim) :type autoregressive_nn: nn.Module :param count_degree: The degree of the polynomial to use for each element-wise transformation. :type count_degree: int :param count_sum: The number of polynomials to sum in each element-wise transformation. :type count_sum: int References: Sum-of-squares polynomial flow. [arXiv:1905.02325] Priyank Jaini, Kira A. Shelby, Yaoliang Yu """ domain = constraints.real codomain = constraints.real bijective = True event_dim = 1 autoregressive = True def __init__(self, autoregressive_nn, input_dim, count_degree, count_sum): super(PolynomialFlow, self).__init__(cache_size=1) self.arn = autoregressive_nn self.input_dim = input_dim self.count_degree = count_degree self.count_sum = count_sum self._cached_logDetJ = None self.c = nn.Parameter(torch.Tensor(input_dim)) self.reset_parameters() # Vector of powers of input dimension powers = torch.arange(1, count_degree + 2, dtype=torch.get_default_dtype()) self.register_buffer('powers', powers) # Build mask of constants mask = self.powers + torch.arange(count_degree+1).unsqueeze(-1).type_as(powers) power_mask = mask mask = mask.reciprocal() self.register_buffer('power_mask', power_mask) self.register_buffer('mask', mask)
[docs] def reset_parameters(self): stdv = 1. / math.sqrt(self.c.size(0)) self.c.data.uniform_(-stdv, stdv)
def _call(self, x): """ :param x: the input into the bijection :type x: torch.Tensor Invokes the bijection x=>y; in the prototypical context of a TransformedDistribution x is a sample from the base distribution (or the output of a previous flow) """ # Calculate the polynomial coefficients # ~ (batch_size, count_sum, count_degree+1, input_dim) A = self.arn(x).view(-1, self.count_sum, self.count_degree + 1, self.input_dim) # Take cross product of coefficients across degree dim # ~ (batch_size, count_sum, count_degree+1, count_degree+1, input_dim) coefs = A.unsqueeze(-2) * A.unsqueeze(-3) # Calculate output as sum-of-squares polynomial x_view = x.view(-1, 1, 1, self.input_dim) x_pow_matrix = x_view.pow(self.power_mask.unsqueeze(-1)).unsqueeze(-4) # Eq (8) from the paper, expanding the squared term and integrating # NOTE: The view_as is necessary because the batch dimensions were collapsed previously y = self.c + (coefs * x_pow_matrix * self.mask.unsqueeze(-1)).sum((1, 2, 3)).view_as(x) # log(|det(J)|) is calculated by the fundamental theorem of calculus, i.e. remove the constant # term and the integral from eq (8) (the equation for this isn't given in the paper) x_pow_matrix = x_view.pow(self.power_mask.unsqueeze(-1) - 1).unsqueeze(-4) self._cached_logDetJ = torch.log((coefs * x_pow_matrix).sum((1, 2, 3)).view_as(x) + 1e-8).sum(-1) return y def _inverse(self, y): """ :param y: the output of the bijection :type y: torch.Tensor Inverts y => x. As noted above, this implementation is incapable of inverting arbitrary values y; rather it assumes y is the result of a previously computed application of the bijector to some x (which was cached on the forward call) """ raise KeyError("PolynomialFlow expected to find key in intermediates cache but didn't")
[docs] def log_abs_det_jacobian(self, x, y): """ Calculates the elementwise determinant of the log Jacobian """ return self._cached_logDetJ