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mvn_diag_plus_low_rank.py
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mvn_diag_plus_low_rank.py
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# Copyright 2016 The TensorFlow Authors. All Rights Reserved.
#
# Licensed under the Apache License, Version 2.0 (the "License");
# you may not use this file except in compliance with the License.
# You may obtain a copy of the License at
#
# http://www.apache.org/licenses/LICENSE-2.0
#
# Unless required by applicable law or agreed to in writing, software
# distributed under the License is distributed on an "AS IS" BASIS,
# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
# See the License for the specific language governing permissions and
# limitations under the License.
# ==============================================================================
"""Multivariate Normal distribution classes."""
from __future__ import absolute_import
from __future__ import division
from __future__ import print_function
from tensorflow.contrib import linalg
from tensorflow.contrib.distributions.python.ops import distribution_util
from tensorflow.contrib.distributions.python.ops import mvn_linear_operator as mvn_linop
from tensorflow.python.framework import ops
__all__ = [
"MultivariateNormalDiagPlusLowRank",
]
class MultivariateNormalDiagPlusLowRank(
mvn_linop.MultivariateNormalLinearOperator):
"""The multivariate normal distribution on `R^k`.
The Multivariate Normal distribution is defined over `R^k` and parameterized
by a (batch of) length-`k` `loc` vector (aka "mu") and a (batch of) `k x k`
`scale` matrix; `covariance = scale @ scale.T` where `@` denotes
matrix-multiplication.
#### Mathematical Details
The probability density function (pdf) is,
```none
pdf(x; loc, scale) = exp(-0.5 ||y||**2) / Z,
y = inv(scale) @ (x - loc),
Z = (2 pi)**(0.5 k) |det(scale)|,
```
where:
* `loc` is a vector in `R^k`,
* `scale` is a linear operator in `R^{k x k}`, `cov = scale @ scale.T`,
* `Z` denotes the normalization constant, and,
* `||y||**2` denotes the squared Euclidean norm of `y`.
A (non-batch) `scale` matrix is:
```none
scale = diag(scale_diag + scale_identity_multiplier ones(k)) +
scale_perturb_factor @ diag(scale_perturb_diag) @ scale_perturb_factor.T
```
where:
* `scale_diag.shape = [k]`,
* `scale_identity_multiplier.shape = []`,
* `scale_perturb_factor.shape = [k, r]`, typically `k >> r`, and,
* `scale_perturb_diag.shape = [r]`.
Additional leading dimensions (if any) will index batches.
If both `scale_diag` and `scale_identity_multiplier` are `None`, then
`scale` is the Identity matrix.
The MultivariateNormal distribution is a member of the [location-scale
family](https://en.wikipedia.org/wiki/Location-scale_family), i.e., it can be
constructed as,
```none
X ~ MultivariateNormal(loc=0, scale=1) # Identity scale, zero shift.
Y = scale @ X + loc
```
#### Examples
```python
tfd = tf.contrib.distributions
# Initialize a single 3-variate Gaussian with covariance `cov = S @ S.T`,
# `S = diag(d) + U @ diag(m) @ U.T`. The perturbation, `U @ diag(m) @ U.T`, is
# a rank-2 update.
mu = [-0.5., 0, 0.5] # shape: [3]
d = [1.5, 0.5, 2] # shape: [3]
U = [[1., 2],
[-1, 1],
[2, -0.5]] # shape: [3, 2]
m = [4., 5] # shape: [2]
mvn = tfd.MultivariateNormalDiagPlusLowRank(
loc=mu
scale_diag=d
scale_perturb_factor=U,
scale_perturb_diag=m)
# Evaluate this on an observation in `R^3`, returning a scalar.
mvn.prob([-1, 0, 1]).eval() # shape: []
# Initialize a 2-batch of 3-variate Gaussians; `S = diag(d) + U @ U.T`.
mu = [[1., 2, 3],
[11, 22, 33]] # shape: [b, k] = [2, 3]
U = [[[1., 2],
[3, 4],
[5, 6]],
[[0.5, 0.75],
[1,0, 0.25],
[1.5, 1.25]]] # shape: [b, k, r] = [2, 3, 2]
m = [[0.1, 0.2],
[0.4, 0.5]] # shape: [b, r] = [2, 2]
mvn = tfd.MultivariateNormalDiagPlusLowRank(
loc=mu,
scale_perturb_factor=U,
scale_perturb_diag=m)
mvn.covariance().eval() # shape: [2, 3, 3]
# ==> [[[ 15.63 31.57 48.51]
# [ 31.57 69.31 105.05]
# [ 48.51 105.05 162.59]]
#
# [[ 2.59 1.41 3.35]
# [ 1.41 2.71 3.34]
# [ 3.35 3.34 8.35]]]
# Compute the pdf of two `R^3` observations (one from each batch);
# return a length-2 vector.
x = [[-0.9, 0, 0.1],
[-10, 0, 9]] # shape: [2, 3]
mvn.prob(x).eval() # shape: [2]
```
"""
def __init__(self,
loc=None,
scale_diag=None,
scale_identity_multiplier=None,
scale_perturb_factor=None,
scale_perturb_diag=None,
validate_args=False,
allow_nan_stats=True,
name="MultivariateNormalDiagPlusLowRank"):
"""Construct Multivariate Normal distribution on `R^k`.
The `batch_shape` is the broadcast shape between `loc` and `scale`
arguments.
The `event_shape` is given by last dimension of the matrix implied by
`scale`. The last dimension of `loc` (if provided) must broadcast with this.
Recall that `covariance = scale @ scale.T`. A (non-batch) `scale` matrix is:
```none
scale = diag(scale_diag + scale_identity_multiplier ones(k)) +
scale_perturb_factor @ diag(scale_perturb_diag) @ scale_perturb_factor.T
```
where:
* `scale_diag.shape = [k]`,
* `scale_identity_multiplier.shape = []`,
* `scale_perturb_factor.shape = [k, r]`, typically `k >> r`, and,
* `scale_perturb_diag.shape = [r]`.
Additional leading dimensions (if any) will index batches.
If both `scale_diag` and `scale_identity_multiplier` are `None`, then
`scale` is the Identity matrix.
Args:
loc: Floating-point `Tensor`. If this is set to `None`, `loc` is
implicitly `0`. When specified, may have shape `[B1, ..., Bb, k]` where
`b >= 0` and `k` is the event size.
scale_diag: Non-zero, floating-point `Tensor` representing a diagonal
matrix added to `scale`. May have shape `[B1, ..., Bb, k]`, `b >= 0`,
and characterizes `b`-batches of `k x k` diagonal matrices added to
`scale`. When both `scale_identity_multiplier` and `scale_diag` are
`None` then `scale` is the `Identity`.
scale_identity_multiplier: Non-zero, floating-point `Tensor` representing
a scaled-identity-matrix added to `scale`. May have shape
`[B1, ..., Bb]`, `b >= 0`, and characterizes `b`-batches of scaled
`k x k` identity matrices added to `scale`. When both
`scale_identity_multiplier` and `scale_diag` are `None` then `scale` is
the `Identity`.
scale_perturb_factor: Floating-point `Tensor` representing a rank-`r`
perturbation added to `scale`. May have shape `[B1, ..., Bb, k, r]`,
`b >= 0`, and characterizes `b`-batches of rank-`r` updates to `scale`.
When `None`, no rank-`r` update is added to `scale`.
scale_perturb_diag: Floating-point `Tensor` representing a diagonal matrix
inside the rank-`r` perturbation added to `scale`. May have shape
`[B1, ..., Bb, r]`, `b >= 0`, and characterizes `b`-batches of `r x r`
diagonal matrices inside the perturbation added to `scale`. When
`None`, an identity matrix is used inside the perturbation. Can only be
specified if `scale_perturb_factor` is also specified.
validate_args: Python `bool`, default `False`. When `True` distribution
parameters are checked for validity despite possibly degrading runtime
performance. When `False` invalid inputs may silently render incorrect
outputs.
allow_nan_stats: Python `bool`, default `True`. When `True`,
statistics (e.g., mean, mode, variance) use the value "`NaN`" to
indicate the result is undefined. When `False`, an exception is raised
if one or more of the statistic's batch members are undefined.
name: Python `str` name prefixed to Ops created by this class.
Raises:
ValueError: if at most `scale_identity_multiplier` is specified.
"""
parameters = locals()
def _convert_to_tensor(x, name):
return None if x is None else ops.convert_to_tensor(x, name=name)
with ops.name_scope(name):
with ops.name_scope("init", values=[
loc, scale_diag, scale_identity_multiplier, scale_perturb_factor,
scale_perturb_diag]):
has_low_rank = (scale_perturb_factor is not None or
scale_perturb_diag is not None)
scale = distribution_util.make_diag_scale(
loc=loc,
scale_diag=scale_diag,
scale_identity_multiplier=scale_identity_multiplier,
validate_args=validate_args,
assert_positive=has_low_rank)
scale_perturb_factor = _convert_to_tensor(
scale_perturb_factor,
name="scale_perturb_factor")
scale_perturb_diag = _convert_to_tensor(
scale_perturb_diag,
name="scale_perturb_diag")
if has_low_rank:
scale = linalg.LinearOperatorLowRankUpdate(
scale,
u=scale_perturb_factor,
diag_update=scale_perturb_diag,
is_diag_update_positive=scale_perturb_diag is None,
is_non_singular=True, # Implied by is_positive_definite=True.
is_self_adjoint=True,
is_positive_definite=True,
is_square=True)
super(MultivariateNormalDiagPlusLowRank, self).__init__(
loc=loc,
scale=scale,
validate_args=validate_args,
allow_nan_stats=allow_nan_stats,
name=name)
self._parameters = parameters