# Source code for pyro.distributions.transforms.polynomial

# Copyright (c) 2017-2019 Uber Technologies, Inc.

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
from pyro.nn import AutoRegressiveNN

[docs]@copy_docs_from(TransformModule)
class Polynomial(TransformModule):
r"""
An autoregressive bijective transform as described in Jaini et al. (2019)
applying following equation element-wise,

: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 :class:~pyro.distributions.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))
>>> param_dims = [(count_degree + 1)*count_sum]
>>> arn = AutoRegressiveNN(input_dim, [input_dim*10], param_dims)
>>> transform = Polynomial(arn, input_dim=input_dim, count_degree=count_degree,
... count_sum=count_sum)
>>> pyro.module("my_transform", transform)  # doctest: +SKIP
>>> flow_dist = dist.TransformedDistribution(base_dist, [transform])
>>> flow_dist.sample()  # doctest: +SKIP

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 a polynomial transform 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:

 Priyank Jaini, Kira A. Shelby, Yaoliang Yu. Sum-of-squares polynomial flow.
[arXiv:1905.02325]

"""

domain = constraints.real_vector
codomain = constraints.real_vector
bijective = True
event_dim = 1
autoregressive = True

def __init__(self, autoregressive_nn, input_dim, count_degree, count_sum):
super().__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)

mask = self.powers + torch.arange(count_degree + 1).unsqueeze(-1).type_as(powers)

[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
:class:~pyro.distributions.TransformedDistribution x is a sample from
the base distribution (or the output of a previous transform)
"""
# 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)

# 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)
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("Polynomial object 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
"""
x_old, y_old = self._cached_x_y
if x is not x_old or y is not y_old:
# This call to the parent class Transform will update the cache
# as well as calling self._call and recalculating y and log_detJ
self(x)

return self._cached_logDetJ

[docs]def polynomial(input_dim, hidden_dims=None):
"""
A helper function to create a :class:~pyro.distributions.transforms.Polynomial
object that takes care of constructing an autoregressive network with the
correct input/output dimensions.

:param input_dim: Dimension of input variable
:type input_dim: int
:param hidden_dims: The desired hidden dimensions of of the autoregressive
network. Defaults to using [input_dim * 10]

"""

count_degree = 4
count_sum = 3
if hidden_dims is None:
hidden_dims = [input_dim * 10]
arn = AutoRegressiveNN(input_dim, hidden_dims, param_dims=[(count_degree + 1) * count_sum])
return Polynomial(arn, input_dim=input_dim, count_degree=count_degree, count_sum=count_sum)