import math
import torch
from torch.autograd import Function
from torch.autograd.function import once_differentiable
from torch.distributions import constraints, Categorical
from pyro.distributions.torch_distribution import TorchDistribution
from pyro.distributions.util import sum_leftmost
[docs]class MixtureOfDiagNormals(TorchDistribution):
"""
Mixture of Normal distributions with arbitrary means and arbitrary
diagonal covariance matrices.
That is, this distribution is a mixture with K components, where each
component distribution is a D-dimensional Normal distribution with a
D-dimensional mean parameter and a D-dimensional diagonal covariance
matrix. The K different component means are gathered into the K x D
dimensional parameter `locs` and the K different scale parameters are
gathered into the K x D dimensional parameter `coord_scale`. The mixture
weights are controlled by a K-dimensional vector of softmax logits,
`component_logits`. This distribution implements pathwise derivatives
for samples from the distribution.
See reference [1] for details on the implementations of the pathwise
derivative. Please consider citing this reference if you use the pathwise
derivative in your research. Note that this distribution does not support
dimension D = 1.
[1] Pathwise Derivatives for Multivariate Distributions, Martin Jankowiak &
Theofanis Karaletsos. arXiv:1806.01856
:param torch.Tensor locs: K x D mean matrix
:param torch.Tensor coord_scale: K x D scale matrix
:param torch.Tensor component_logits: K-dimensional vector of softmax logits
"""
has_rsample = True
arg_constraints = {"locs": constraints.real, "coord_scale": constraints.positive,
"component_logits": constraints.real}
def __init__(self, locs, coord_scale, component_logits):
self.batch_mode = (locs.dim() > 2)
assert(coord_scale.shape == locs.shape)
assert(self.batch_mode or locs.dim() == 2), \
"The locs parameter in MixtureOfDiagNormals should be K x D dimensional (or B x K x D if doing batches)"
if not self.batch_mode:
assert(coord_scale.dim() == 2), \
"The coord_scale parameter in MixtureOfDiagNormals should be K x D dimensional"
assert(component_logits.dim() == 1), \
"The component_logits parameter in MixtureOfDiagNormals should be K dimensional"
assert(component_logits.size(-1) == locs.size(-2))
batch_shape = ()
else:
assert(coord_scale.dim() > 2), \
"The coord_scale parameter in MixtureOfDiagNormals should be B x K x D dimensional"
assert(component_logits.dim() > 1), \
"The component_logits parameter in MixtureOfDiagNormals should be B x K dimensional"
assert(component_logits.size(-1) == locs.size(-2))
batch_shape = tuple(locs.shape[:-2])
self.locs = locs
self.coord_scale = coord_scale
self.component_logits = component_logits
self.dim = locs.size(-1)
self.categorical = Categorical(logits=component_logits)
self.probs = self.categorical.probs
super(MixtureOfDiagNormals, self).__init__(batch_shape=torch.Size(batch_shape),
event_shape=torch.Size((self.dim,)))
[docs] def expand(self, batch_shape, _instance=None):
new = self._get_checked_instance(MixtureOfDiagNormals, _instance)
new.batch_mode = True
batch_shape = torch.Size(batch_shape)
new.dim = self.dim
new.locs = self.locs.expand(batch_shape + self.locs.shape[-2:])
new.coord_scale = self.coord_scale.expand(batch_shape + self.coord_scale.shape[-2:])
new.component_logits = self.component_logits.expand(batch_shape + self.component_logits.shape[-1:])
new.categorical = self.categorical.expand(batch_shape)
new.probs = self.probs.expand(batch_shape + self.probs.shape[-1:])
super(MixtureOfDiagNormals, new).__init__(batch_shape, self.event_shape, validate_args=False)
new._validate_args = self._validate_args
return new
[docs] def log_prob(self, value):
epsilon = (value.unsqueeze(-2) - self.locs) / self.coord_scale # L B K D
eps_sqr = 0.5 * torch.pow(epsilon, 2.0).sum(-1) # L B K
eps_sqr_min = torch.min(eps_sqr, -1)[0] # L B K
coord_scale_prod_log_sum = self.coord_scale.log().sum(-1) # B K
result = self.categorical.logits + (-eps_sqr + eps_sqr_min.unsqueeze(-1)) - coord_scale_prod_log_sum # L B K
result = torch.logsumexp(result, dim=-1) # L B
result = result - 0.5 * math.log(2.0 * math.pi) * float(self.dim)
result = result - eps_sqr_min
return result
[docs] def rsample(self, sample_shape=torch.Size()):
which = self.categorical.sample(sample_shape)
return _MixDiagNormalSample.apply(self.locs, self.coord_scale,
self.component_logits, self.categorical.probs, which,
sample_shape + self.locs.shape[:-2] + (self.dim,))
class _MixDiagNormalSample(Function):
@staticmethod
def forward(ctx, locs, scales, component_logits, pis, which, noise_shape):
dim = scales.size(-1)
white = locs.new(noise_shape).normal_()
n_unsqueezes = locs.dim() - which.dim()
for _ in range(n_unsqueezes):
which = which.unsqueeze(-1)
which_expand = which.expand(tuple(which.shape[:-1] + (dim,)))
loc = torch.gather(locs, -2, which_expand).squeeze(-2)
sigma = torch.gather(scales, -2, which_expand).squeeze(-2)
z = loc + sigma * white
ctx.save_for_backward(z, scales, locs, component_logits, pis)
return z
@staticmethod
@once_differentiable
def backward(ctx, grad_output):
z, scales, locs, logits, pis = ctx.saved_tensors
dim = scales.size(-1)
K = logits.size(-1)
g = grad_output # l b i
g = g.unsqueeze(-2) # l b 1 i
batch_dims = locs.dim() - 2
locs_tilde = locs / scales # b j i
sigma_0 = torch.min(scales, -2, keepdim=True)[0] # b 1 i
z_shift = (z.unsqueeze(-2) - locs) / sigma_0 # l b j i
z_tilde = z.unsqueeze(-2) / scales - locs_tilde # l b j i
mu_cd = locs.unsqueeze(-2) - locs.unsqueeze(-3) # b c d i
mu_cd_norm = torch.pow(mu_cd, 2.0).sum(-1).sqrt() # b c d
mu_cd /= mu_cd_norm.unsqueeze(-1) # b c d i
diagonals = torch.empty((K,), dtype=torch.long, device=z.device)
torch.arange(K, out=diagonals)
mu_cd[..., diagonals, diagonals, :] = 0.0
mu_ll_cd = (locs.unsqueeze(-2) * mu_cd).sum(-1) # b c d
z_ll_cd = (z.unsqueeze(-2).unsqueeze(-2) * mu_cd).sum(-1) # l b c d
z_perp_cd = z.unsqueeze(-2).unsqueeze(-2) - z_ll_cd.unsqueeze(-1) * mu_cd # l b c d i
z_perp_cd_sqr = torch.pow(z_perp_cd, 2.0).sum(-1) # l b c d
shift_indices = torch.empty((dim,), dtype=torch.long, device=z.device)
torch.arange(dim, out=shift_indices)
shift_indices = shift_indices - 1
shift_indices[0] = 0
z_shift_cumsum = torch.pow(z_shift, 2.0)
z_shift_cumsum = z_shift_cumsum.sum(-1, keepdim=True) - torch.cumsum(z_shift_cumsum, dim=-1) # l b j i
z_tilde_cumsum = torch.cumsum(torch.pow(z_tilde, 2.0), dim=-1) # l b j i
z_tilde_cumsum = torch.index_select(z_tilde_cumsum, -1, shift_indices)
z_tilde_cumsum[..., 0] = 0.0
r_sqr_ji = z_shift_cumsum + z_tilde_cumsum # l b j i
log_scales = torch.log(scales) # b j i
epsilons_sqr = torch.pow(z_tilde, 2.0) # l b j i
log_qs = -0.5 * epsilons_sqr - 0.5 * math.log(2.0 * math.pi) - log_scales # l b j i
log_q_j = log_qs.sum(-1, keepdim=True) # l b j 1
q_j = torch.exp(log_q_j) # l b j 1
q_tot = (pis * q_j.squeeze(-1)).sum(-1) # l b
q_tot = q_tot.unsqueeze(-1) # l b 1
root_two = math.sqrt(2.0)
shift_log_scales = log_scales[..., shift_indices]
shift_log_scales[..., 0] = 0.0
sigma_products = torch.cumsum(shift_log_scales, dim=-1).exp() # b j i
reverse_indices = torch.tensor(range(dim - 1, -1, -1), dtype=torch.long, device=z.device)
reverse_log_sigma_0 = sigma_0.log()[..., reverse_indices] # b 1 i
sigma_0_products = torch.cumsum(reverse_log_sigma_0, dim=-1).exp()[..., reverse_indices - 1] # b 1 i
sigma_0_products[..., -1] = 1.0
sigma_products *= sigma_0_products
logits_grad = torch.erf(z_tilde / root_two) - torch.erf(z_shift / root_two) # l b j i
logits_grad *= torch.exp(-0.5 * r_sqr_ji) # l b j i
logits_grad = (logits_grad * g / sigma_products).sum(-1) # l b j
logits_grad = sum_leftmost(logits_grad / q_tot, -1 - batch_dims) # b j
logits_grad *= 0.5 * math.pow(2.0 * math.pi, -0.5 * (dim - 1))
logits_grad = -pis * logits_grad
logits_grad = logits_grad - logits_grad.sum(-1, keepdim=True) * pis
mu_ll_dc = torch.transpose(mu_ll_cd, -1, -2)
v_cd = torch.erf((z_ll_cd - mu_ll_cd) / root_two) - torch.erf((z_ll_cd + mu_ll_dc) / root_two)
v_cd *= torch.exp(-0.5 * z_perp_cd_sqr) # l b c d
mu_cd_g = (g.unsqueeze(-2) * mu_cd).sum(-1) # l b c d
v_cd *= -mu_cd_g * pis.unsqueeze(-2) * 0.5 * math.pow(2.0 * math.pi, -0.5 * (dim - 1)) # l b c d
v_cd = pis * sum_leftmost(v_cd.sum(-1) / q_tot, -1 - batch_dims)
logits_grad += v_cd
prefactor = pis.unsqueeze(-1) * q_j * g / q_tot.unsqueeze(-1)
locs_grad = sum_leftmost(prefactor, -2 - batch_dims)
scales_grad = sum_leftmost(prefactor * z_tilde, -2 - batch_dims)
return locs_grad, scales_grad, logits_grad, None, None, None