#!/usr/bin/env python3
from __future__ import annotations
import math
import warnings
from numbers import Number
from typing import Optional, Tuple, Union
import torch
from linear_operator import to_dense, to_linear_operator
from linear_operator.operators import DiagLinearOperator, LinearOperator, RootLinearOperator
from torch import Tensor
from torch.distributions import MultivariateNormal as TMultivariateNormal
from torch.distributions.kl import register_kl
from torch.distributions.utils import _standard_normal, lazy_property
from .. import settings
from ..utils.warnings import NumericalWarning
from .distribution import Distribution
[docs]class MultivariateNormal(TMultivariateNormal, Distribution):
"""
Constructs a multivariate normal random variable, based on mean and covariance.
Can be multivariate, or a batch of multivariate normals
Passing a vector mean corresponds to a multivariate normal.
Passing a matrix mean corresponds to a batch of multivariate normals.
:param mean: `... x N` mean of mvn distribution.
:param covariance_matrix: `... x N X N` covariance matrix of mvn distribution.
:param validate_args: If True, validate `mean` anad `covariance_matrix` arguments. (Default: False.)
:ivar torch.Size base_sample_shape: The shape of a base sample (without
batching) that is used to generate a single sample.
:ivar torch.Tensor covariance_matrix: The covariance matrix, represented as a dense :class:`torch.Tensor`
:ivar ~linear_operator.LinearOperator lazy_covariance_matrix: The covariance matrix, represented
as a :class:`~linear_operator.LinearOperator`.
:ivar torch.Tensor mean: The mean.
:ivar torch.Tensor stddev: The standard deviation.
:ivar torch.Tensor variance: The variance.
"""
def __init__(self, mean: Tensor, covariance_matrix: Union[Tensor, LinearOperator], validate_args: bool = False):
self._islazy = isinstance(mean, LinearOperator) or isinstance(covariance_matrix, LinearOperator)
if self._islazy:
if validate_args:
ms = mean.size(-1)
cs1 = covariance_matrix.size(-1)
cs2 = covariance_matrix.size(-2)
if not (ms == cs1 and ms == cs2):
raise ValueError(f"Wrong shapes in {self._repr_sizes(mean, covariance_matrix)}")
self.loc = mean
self._covar = covariance_matrix
self.__unbroadcasted_scale_tril = None
self._validate_args = validate_args
batch_shape = torch.broadcast_shapes(self.loc.shape[:-1], covariance_matrix.shape[:-2])
event_shape = self.loc.shape[-1:]
# TODO: Integrate argument validation for LinearOperators into torch.distribution validation logic
super(TMultivariateNormal, self).__init__(batch_shape, event_shape, validate_args=False)
else:
super().__init__(loc=mean, covariance_matrix=covariance_matrix, validate_args=validate_args)
def _extended_shape(self, sample_shape: torch.Size = torch.Size()) -> torch.Size:
"""
Returns the size of the sample returned by the distribution, given
a `sample_shape`. Note, that the batch and event shapes of a distribution
instance are fixed at the time of construction. If this is empty, the
returned shape is upcast to (1,).
:param sample_shape: the size of the sample to be drawn.
"""
if not isinstance(sample_shape, torch.Size):
sample_shape = torch.Size(sample_shape)
return sample_shape + self._batch_shape + self.base_sample_shape
@staticmethod
def _repr_sizes(mean: Tensor, covariance_matrix: Union[Tensor, LinearOperator]) -> str:
return f"MultivariateNormal(loc: {mean.size()}, scale: {covariance_matrix.size()})"
@property
def _unbroadcasted_scale_tril(self) -> Tensor:
if self.islazy and self.__unbroadcasted_scale_tril is None:
# cache root decoposition
ust = to_dense(self.lazy_covariance_matrix.cholesky())
self.__unbroadcasted_scale_tril = ust
return self.__unbroadcasted_scale_tril
@_unbroadcasted_scale_tril.setter
def _unbroadcasted_scale_tril(self, ust: Tensor):
if self.islazy:
raise NotImplementedError("Cannot set _unbroadcasted_scale_tril for lazy MVN distributions")
else:
self.__unbroadcasted_scale_tril = ust
[docs] def add_jitter(self, noise: float = 1e-4) -> MultivariateNormal:
r"""
Adds a small constant diagonal to the MVN covariance matrix for numerical stability.
:param noise: The size of the constant diagonal.
"""
return self.__class__(self.mean, self.lazy_covariance_matrix.add_jitter(noise))
@property
def base_sample_shape(self) -> torch.Size:
base_sample_shape = self.event_shape
if isinstance(self.lazy_covariance_matrix, RootLinearOperator):
base_sample_shape = self.lazy_covariance_matrix.root.shape[-1:]
return base_sample_shape
@lazy_property
def covariance_matrix(self) -> Tensor:
if self.islazy:
return self._covar.to_dense()
else:
return super().covariance_matrix
[docs] def confidence_region(self) -> Tuple[Tensor, Tensor]:
"""
Returns 2 standard deviations above and below the mean.
:return: Pair of tensors of size `... x N`, where N is the
dimensionality of the random variable. The first (second) Tensor is the
lower (upper) end of the confidence region.
"""
std2 = self.stddev.mul_(2)
mean = self.mean
return mean.sub(std2), mean.add(std2)
[docs] def expand(self, batch_size: torch.Size) -> MultivariateNormal:
r"""
See :py:meth:`torch.distributions.Distribution.expand
<torch.distributions.distribution.Distribution.expand>`.
"""
new_loc = self.loc.expand(torch.Size(batch_size) + self.loc.shape[-1:])
new_covar = self._covar.expand(torch.Size(batch_size) + self._covar.shape[-2:])
res = self.__class__(new_loc, new_covar)
return res
[docs] def get_base_samples(self, sample_shape: torch.Size = torch.Size()) -> Tensor:
r"""
Returns i.i.d. standard Normal samples to be used with
:py:meth:`MultivariateNormal.rsample(base_samples=base_samples)
<gpytorch.distributions.MultivariateNormal.rsample>`.
:param sample_shape: The number of samples to generate. (Default: `torch.Size([])`.)
:return: A `*sample_shape x *batch_shape x N` tensor of i.i.d. standard Normal samples.
"""
with torch.no_grad():
shape = self._extended_shape(sample_shape)
base_samples = _standard_normal(shape, dtype=self.loc.dtype, device=self.loc.device)
return base_samples
@lazy_property
def lazy_covariance_matrix(self) -> LinearOperator:
if self.islazy:
return self._covar
else:
return to_linear_operator(super().covariance_matrix)
[docs] def log_prob(self, value: Tensor) -> Tensor:
r"""
See :py:meth:`torch.distributions.Distribution.log_prob
<torch.distributions.distribution.Distribution.log_prob>`.
"""
if settings.fast_computations.log_prob.off():
return super().log_prob(value)
if self._validate_args:
self._validate_sample(value)
mean, covar = self.loc, self.lazy_covariance_matrix
diff = value - mean
# Repeat the covar to match the batch shape of diff
if diff.shape[:-1] != covar.batch_shape:
if len(diff.shape[:-1]) < len(covar.batch_shape):
diff = diff.expand(covar.shape[:-1])
else:
padded_batch_shape = (*(1 for _ in range(diff.dim() + 1 - covar.dim())), *covar.batch_shape)
covar = covar.repeat(
*(diff_size // covar_size for diff_size, covar_size in zip(diff.shape[:-1], padded_batch_shape)),
1,
1,
)
# Get log determininant and first part of quadratic form
covar = covar.evaluate_kernel()
inv_quad, logdet = covar.inv_quad_logdet(inv_quad_rhs=diff.unsqueeze(-1), logdet=True)
res = -0.5 * sum([inv_quad, logdet, diff.size(-1) * math.log(2 * math.pi)])
return res
[docs] def rsample(self, sample_shape: torch.Size = torch.Size(), base_samples: Optional[Tensor] = None) -> Tensor:
r"""
Generates a `sample_shape` shaped reparameterized sample or `sample_shape`
shaped batch of reparameterized samples if the distribution parameters
are batched.
For the MultivariateNormal distribution, this is accomplished through:
.. math::
\boldsymbol \mu + \mathbf L \boldsymbol \epsilon
where :math:`\boldsymbol \mu \in \mathcal R^N` is the MVN mean,
:math:`\mathbf L \in \mathcal R^{N \times N}` is a "root" of the
covariance matrix :math:`\mathbf K` (i.e. :math:`\mathbf L \mathbf
L^\top = \mathbf K`), and :math:`\boldsymbol \epsilon \in \mathcal R^N` is a
vector of (approximately) i.i.d. standard Normal random variables.
:param sample_shape: The number of samples to generate. (Default: `torch.Size([])`.)
:param base_samples: The `*sample_shape x *batch_shape x N` tensor of
i.i.d. (or approximately i.i.d.) standard Normal samples to
reparameterize. (Default: None.)
:return: A `*sample_shape x *batch_shape x N` tensor of i.i.d. reparameterized samples.
"""
covar = self.lazy_covariance_matrix
if base_samples is None:
# Create some samples
num_samples = sample_shape.numel() or 1
# Get samples
res = covar.zero_mean_mvn_samples(num_samples) + self.loc.unsqueeze(0)
res = res.view(sample_shape + self.loc.shape)
else:
covar_root = covar.root_decomposition().root
# Make sure that the base samples agree with the distribution
if (
self.loc.shape != base_samples.shape[-self.loc.dim() :]
and covar_root.shape[-1] < base_samples.shape[-1]
):
raise RuntimeError(
"The size of base_samples (minus sample shape dimensions) should agree with the size "
"of self.loc. Expected ...{} but got {}".format(self.loc.shape, base_samples.shape)
)
# Determine what the appropriate sample_shape parameter is
sample_shape = base_samples.shape[: base_samples.dim() - self.loc.dim()]
# Reshape samples to be batch_size x num_dim x num_samples
# or num_bim x num_samples
base_samples = base_samples.view(-1, *self.loc.shape[:-1], covar_root.shape[-1])
base_samples = base_samples.permute(*range(1, self.loc.dim() + 1), 0)
# Now reparameterize those base samples
# If necessary, adjust base_samples for rank of root decomposition
if covar_root.shape[-1] < base_samples.shape[-2]:
base_samples = base_samples[..., : covar_root.shape[-1], :]
elif covar_root.shape[-1] > base_samples.shape[-2]:
# raise RuntimeError("Incompatible dimension of `base_samples`")
covar_root = covar_root.transpose(-2, -1)
res = covar_root.matmul(base_samples) + self.loc.unsqueeze(-1)
# Permute and reshape new samples to be original size
res = res.permute(-1, *range(self.loc.dim())).contiguous()
res = res.view(sample_shape + self.loc.shape)
return res
[docs] def sample(self, sample_shape: torch.Size = torch.Size(), base_samples: Optional[Tensor] = None) -> Tensor:
r"""
Generates a `sample_shape` shaped sample or `sample_shape`
shaped batch of samples if the distribution parameters
are batched.
Note that these samples are not reparameterized and therefore cannot be backpropagated through.
:param sample_shape: The number of samples to generate. (Default: `torch.Size([])`.)
:param base_samples: The `*sample_shape x *batch_shape x N` tensor of
i.i.d. (or approximately i.i.d.) standard Normal samples to
reparameterize. (Default: None.)
:return: A `*sample_shape x *batch_shape x N` tensor of i.i.d. samples.
"""
with torch.no_grad():
return self.rsample(sample_shape=sample_shape, base_samples=base_samples)
@property
def stddev(self) -> Tensor:
# self.variance is guaranteed to be positive, because we do clamping.
return self.variance.sqrt()
[docs] def to_data_independent_dist(self) -> torch.distributions.Normal:
"""
Convert a `... x N` MVN distribution into a batch of independent Normal distributions.
Essentially, this throws away all covariance information
and treats all dimensions as batch dimensions.
:returns: A (data-independent) Normal distribution with batch shape `*batch_shape x N`.
"""
# Create batch distribution where all data are independent, but the tasks are dependent
try:
# If pyro is installed, use that set of base distributions
import pyro.distributions as base_distributions
except ImportError:
# Otherwise, use PyTorch
import torch.distributions as base_distributions
return base_distributions.Normal(self.mean, self.stddev)
@property
def variance(self) -> Tensor:
if self.islazy:
# overwrite this since torch MVN uses unbroadcasted_scale_tril for this
diag = self.lazy_covariance_matrix.diagonal(dim1=-1, dim2=-2)
diag = diag.view(diag.shape[:-1] + self._event_shape)
variance = diag.expand(self._batch_shape + self._event_shape)
else:
variance = super().variance
# Check to make sure that variance isn't lower than minimum allowed value (default 1e-6).
# This ensures that all variances are positive
min_variance = settings.min_variance.value(variance.dtype)
if variance.lt(min_variance).any():
warnings.warn(
f"Negative variance values detected. "
"This is likely due to numerical instabilities. "
f"Rounding negative variances up to {min_variance}.",
NumericalWarning,
)
variance = variance.clamp_min(min_variance)
return variance
def __add__(self, other: MultivariateNormal) -> MultivariateNormal:
if isinstance(other, MultivariateNormal):
return self.__class__(
mean=self.mean + other.mean,
covariance_matrix=(self.lazy_covariance_matrix + other.lazy_covariance_matrix),
)
elif isinstance(other, int) or isinstance(other, float):
return self.__class__(self.mean + other, self.lazy_covariance_matrix)
else:
raise RuntimeError("Unsupported type {} for addition w/ MultivariateNormal".format(type(other)))
[docs] def __getitem__(self, idx) -> MultivariateNormal:
r"""
Constructs a new MultivariateNormal that represents a random variable
modified by an indexing operation.
The mean and covariance matrix arguments are indexed accordingly.
:param idx: Index to apply.
"""
if not isinstance(idx, tuple):
idx = (idx,)
rest_idx = idx[:-1]
last_idx = idx[-1]
new_mean = self.mean[idx]
if len(idx) <= self.mean.dim() - 1 and (Ellipsis not in rest_idx):
new_cov = self.lazy_covariance_matrix[idx]
elif len(idx) > self.mean.dim():
raise IndexError(f"Index {idx} has too many dimensions")
else:
# In this case we know last_idx corresponds to the last dimension
# of mean and the last two dimensions of lazy_covariance_matrix
if isinstance(last_idx, int):
new_cov = DiagLinearOperator(
self.lazy_covariance_matrix.diagonal(dim1=-1, dim2=-2)[(*rest_idx, last_idx)]
)
elif isinstance(last_idx, slice):
new_cov = self.lazy_covariance_matrix[(*rest_idx, last_idx, last_idx)]
elif last_idx is (...):
new_cov = self.lazy_covariance_matrix[rest_idx]
else:
new_cov = self.lazy_covariance_matrix[(*rest_idx, last_idx, slice(None, None, None))][..., last_idx]
return self.__class__(mean=new_mean, covariance_matrix=new_cov)
def __mul__(self, other: Number) -> MultivariateNormal:
if not (isinstance(other, int) or isinstance(other, float)):
raise RuntimeError("Can only multiply by scalars")
if other == 1:
return self
return self.__class__(mean=self.mean * other, covariance_matrix=self.lazy_covariance_matrix * (other**2))
def __radd__(self, other: MultivariateNormal) -> MultivariateNormal:
if other == 0:
return self
return self.__add__(other)
def __truediv__(self, other: Number) -> MultivariateNormal:
return self.__mul__(1.0 / other)
@register_kl(MultivariateNormal, MultivariateNormal)
def kl_mvn_mvn(p_dist: MultivariateNormal, q_dist: MultivariateNormal) -> Tensor:
output_shape = torch.broadcast_shapes(p_dist.batch_shape, q_dist.batch_shape)
if output_shape != p_dist.batch_shape:
p_dist = p_dist.expand(output_shape)
if output_shape != q_dist.batch_shape:
q_dist = q_dist.expand(output_shape)
q_mean = q_dist.loc
q_covar = q_dist.lazy_covariance_matrix
p_mean = p_dist.loc
p_covar = p_dist.lazy_covariance_matrix
root_p_covar = p_covar.root_decomposition().root.to_dense()
mean_diffs = p_mean - q_mean
if isinstance(root_p_covar, LinearOperator):
# right now this just catches if root_p_covar is a DiagLinearOperator,
# but we may want to be smarter about this in the future
root_p_covar = root_p_covar.to_dense()
inv_quad_rhs = torch.cat([mean_diffs.unsqueeze(-1), root_p_covar], -1)
logdet_p_covar = p_covar.logdet()
trace_plus_inv_quad_form, logdet_q_covar = q_covar.inv_quad_logdet(inv_quad_rhs=inv_quad_rhs, logdet=True)
# Compute the KL Divergence.
res = 0.5 * sum([logdet_q_covar, logdet_p_covar.mul(-1), trace_plus_inv_quad_form, -float(mean_diffs.size(-1))])
return res