Determinantal point process (DPP) distribution.
Inherits From: AutoCompositeTensorDistribution
, Distribution
oryx.distributions.DeterminantalPointProcess(
eigenvalues,
eigenvectors,
validate_args=False,
allow_nan_stats=False,
name='DeterminantalPointProcess'
)
The DPP disribution parameterized by the eigenvalues and eigenvectors of the Lensemble matrix. The Lensemble matrix indicates the degree of "repulsion" between pairs of items.
Mathematical details
A Determinantal Point Process is a distribution over subsets of n
items,
called the ground set. The DPP is parameterized by a positive definite
matrix of shape n x n
, the Lensemble matrix. It assigns to any subset S
of {1, ..., n}
the probability:
Pr(S) = det(L_S) / det(I + L)
where:
L
is the Lensemble matrix parameterized byeigenvalues
andeigenvectors
, i.e.L = U D U^T
forU = eigenvectors
andD = eigenvalues
.L_S
is the principal submatrix ofL
indexed by items inS
. In Numpy slicing notation,L_S = L[S, :][:, S]
.det
is the matrix determinant.
Marginal probabilities, i.e. the probability that a sample from the DPP contains the subset S, are obtained by way of the marginal kernel:
K = L / (I + L)
where /
is the matrix inverse.
When sampling a random set A
from the DPP, the marginal probability of S
,
given by exp(dpp.marginal_log_prob(S))
, is:
Pr(A is a superset of S) = det(K_S)
This is a marginal probability in the following sense. If we think of the
DPP as a joint distribution over n
binary indicator variables, each telling
whether a given element is in a given subset S
, then we can consider the
marginal distribution obtained by "summing" out some of these binary
indicators. The resulting marginal distribution happens also to be a DPP. What
is referred to as the marginal_log_prob
of S
(under the original DPP) is
just the log_prob
of S
under the marginal DPP, obtained by summing out the
indicators of the complement of S. This tells us the (log) probability that
a sample from the full DPP includes S
as a subset.
Written in terms of sets, with each S'
a subset of the complement of S
:
det(K_S) = sum_{S' s.t. S' intersect S is empty} [ Pr(S union S') ]
where Pr(S union S')
is the probability of sampling exactly S union S'
from the DPP.
For further detail, see Theorem 2.2 of [3].
Repulsion
Rewriting L = B B^T
(which in particular can be done using B = U sqrt(D)
,
where D
are the eigenvalues and U
the eigenvectors), we have
Pr(S) = Vol^2(b_s1, b_s2, ..., b_sk)
where b_s1, ...
is the s1
th column of B
. Hence, the probability of
sampling two points simultaneously decreases as a function of how colinear
their corresponding eigenvectors are.
Sampling
Sampling is implemented following the algorithm introduced in 2, and proceeds in two phases.
Given an orthonormalization L = U D U^T
:
First, an elementary DPP (EDPP) is built by sampling a subset of eigenvectors
S
from a Bernoulli distribution with probs equal toD / (D + 1)
. This EDPP has the same eigenvectorsU
asL
, but its eigenvalues are1
iff the corresponding Bernoulli trial was succesful,0
otherwise.Then, a number of points
k
equal to the number of selected eigenvalues is selected iteratively from the elementary DPP. After sampling a pointi
, the kernel is updated by projecting it onto the subspace of eigenvectors orthogonal to thei
th basis vector.
Examples
Sample points on the unit square grid:
import itertools
from tensorflow_probability.python.internal.backend import jax as tf
from tensorflow_probability.substrates import numpy as tfp
import matplotlib.pyplot as plt
tfd = tfp.distributions
tfpk = tfp.math.psd_kernels
grid_size = 16
# Generate grid_size**2 pts on the unit square.
grid = np.arange(0, 1, 1./grid_size)
points = np.array(list(itertools.product(grid, grid)))
# Create the kernel L that parameterizes the DPP.
kernel_amplitude = 2.
kernel_lengthscale = 2. / grid_size
kernel = tfpk.ExponentiatedQuadratic(kernel_amplitude, kernel_lengthscale)
kernel_matrix = kernel.matrix(points, points)
eigenvalues, eigenvectors = tf.linalg.eigh(kernel_matrix)
dpp = tfd.DeterminantalPointProcess(eigenvalues, eigenvectors)
# The innermost dimension of the result of `dpp.sample` is a multihot
# encoding of a subset of {1, ..., ground_set_size}.
plt.figure(figsize=(6, 6))
for i, samp in enumerate(dpp.sample(4, seed=(1, 2))): # 4 x grid_size**2
plt.subplot(221 + i)
plt.scatter(*points[np.where(samp)].T)
plt.xticks([])
plt.yticks([])
plt.tight_layout()
plt.show()
# Like any TFP distribution, the DPP supports batching and shaped samples.
kernel_amplitude = [2., 3, 4] # Build a batch of 3 PSD kernels.
kernel_lengthscale = 2. / grid_size
kernel = tfpk.ExponentiatedQuadratic(kernel_amplitude, kernel_lengthscale)
kernel_matrix = kernel.matrix(points, points) # 3 x 256 x 256
eigenvalues, eigenvectors = tf.linalg.eigh(kernel_matrix)
dpp = tfd.DeterminantalPointProcess(eigenvalues, eigenvectors)
print(dpp) # batch shape: [3], event shape: [256]
samps = dpp.sample(2, seed=(10, 20))
print(samps.shape) # shape: [2, 3, 256]
print(dpp.log_prob(samps)) # tensor with shape [2, 3]
References
[1]: Odile Macchi. The coincidence approach to stochastic point processes. Advances in Applied Probability, 1975.
[2]: J. Ben Hough, Manjunath Krishnapur, Yuval Peres, Balint Virag. Determinantal point processes and independence. Probability Surveys, 2006. https://arxiv.org/abs/math/0503110
[3]: Alex Kulesza, Ben Taskar. Determinantal point processes for machine learning. Foundations and Trends in Machine Learning, 2012. https://arxiv.org/abs/1207.6083
Methods
batch_shape_tensor
batch_shape_tensor(
name='batch_shape_tensor'
)
Shape of a single sample from a single event index as a 1D Tensor
.
The batch dimensions are indexes into independent, nonidentical parameterizations of this distribution.
Args  

name

name to give to the op 
Returns  

batch_shape

Tensor .

cdf
cdf(
value, name='cdf', **kwargs
)
Cumulative distribution function.
Given random variable X
, the cumulative distribution function cdf
is:
cdf(x) := P[X <= x]
Args  

value

float or double Tensor .

name

Python str prepended to names of ops created by this function.

**kwargs

Named arguments forwarded to subclass implementation. 
Returns  

cdf

a Tensor of shape sample_shape(x) + self.batch_shape with
values of type self.dtype .

copy
copy(
**override_parameters_kwargs
)
Creates a deep copy of the distribution.
Args  

**override_parameters_kwargs

String/value dictionary of initialization arguments to override with new values. 
Returns  

distribution

A new instance of type(self) initialized from the union
of self.parameters and override_parameters_kwargs, i.e.,
dict(self.parameters, **override_parameters_kwargs) .

covariance
covariance(
name='covariance', **kwargs
)
Covariance.
Covariance is (possibly) defined only for nonscalarevent distributions.
For example, for a lengthk
, vectorvalued distribution, it is calculated
as,
Cov[i, j] = Covariance(X_i, X_j) = E[(X_i  E[X_i]) (X_j  E[X_j])]
where Cov
is a (batch of) k x k
matrix, 0 <= (i, j) < k
, and E
denotes expectation.
Alternatively, for nonvector, multivariate distributions (e.g.,
matrixvalued, Wishart), Covariance
shall return a (batch of) matrices
under some vectorization of the events, i.e.,
Cov[i, j] = Covariance(Vec(X)_i, Vec(X)_j) = [as above]
where Cov
is a (batch of) k' x k'
matrices,
0 <= (i, j) < k' = reduce_prod(event_shape)
, and Vec
is some function
mapping indices of this distribution's event dimensions to indices of a
lengthk'
vector.
Args  

name

Python str prepended to names of ops created by this function.

**kwargs

Named arguments forwarded to subclass implementation. 
Returns  

covariance

Floatingpoint Tensor with shape [B1, ..., Bn, k', k']
where the first n dimensions are batch coordinates and
k' = reduce_prod(self.event_shape) .

cross_entropy
cross_entropy(
other, name='cross_entropy'
)
Computes the (Shannon) cross entropy.
Denote this distribution (self
) by P
and the other
distribution by
Q
. Assuming P, Q
are absolutely continuous with respect to
one another and permit densities p(x) dr(x)
and q(x) dr(x)
, (Shannon)
cross entropy is defined as:
H[P, Q] = E_p[log q(X)] = int_F p(x) log q(x) dr(x)
where F
denotes the support of the random variable X ~ P
.
Args  

other

tfp.distributions.Distribution instance.

name

Python str prepended to names of ops created by this function.

Returns  

cross_entropy

self.dtype Tensor with shape [B1, ..., Bn]
representing n different calculations of (Shannon) cross entropy.

entropy
entropy(
name='entropy', **kwargs
)
Shannon entropy in nats.
event_shape_tensor
event_shape_tensor(
name='event_shape_tensor'
)
Shape of a single sample from a single batch as a 1D int32 Tensor
.
Args  

name

name to give to the op 
Returns  

event_shape

Tensor .

experimental_default_event_space_bijector
experimental_default_event_space_bijector(
*args, **kwargs
)
Bijector mapping the reals (R**n) to the event space of the distribution.
Distributions with continuous support may implement
_default_event_space_bijector
which returns a subclass of
tfp.bijectors.Bijector
that maps R**n to the distribution's event space.
For example, the default bijector for the Beta
distribution
is tfp.bijectors.Sigmoid()
, which maps the real line to [0, 1]
, the
support of the Beta
distribution. The default bijector for the
CholeskyLKJ
distribution is tfp.bijectors.CorrelationCholesky
, which
maps R^(k * (k1) // 2) to the submanifold of k x k lower triangular
matrices with ones along the diagonal.
The purpose of experimental_default_event_space_bijector
is
to enable gradient descent in an unconstrained space for Variational
Inference and Hamiltonian Monte Carlo methods. Some effort has been made to
choose bijectors such that the tails of the distribution in the
unconstrained space are between Gaussian and Exponential.
For distributions with discrete event space, or for which TFP currently
lacks a suitable bijector, this function returns None
.
Args  

*args

Passed to implementation _default_event_space_bijector .

**kwargs

Passed to implementation _default_event_space_bijector .

Returns  

event_space_bijector

Bijector instance or None .

experimental_fit
@classmethod
experimental_fit( value, sample_ndims=1, validate_args=False, **init_kwargs )
Instantiates a distribution that maximizes the likelihood of x
.
Args  

value

a Tensor valid sample from this distribution family.

sample_ndims

Positive int Tensor number of leftmost dimensions of
value that index i.i.d. samples.
Default value: 1 .

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.
Default value: False .

**init_kwargs

Additional keyword arguments passed through to
cls.__init__ . These take precedence in case of collision with the
fitted parameters; for example,
tfd.Normal.experimental_fit([1., 1.], scale=20.) returns a Normal
distribution with scale=20. rather than the maximum likelihood
parameter scale=0. .

Returns  

maximum_likelihood_instance

instance of cls with parameters that
maximize the likelihood of value .

experimental_local_measure
experimental_local_measure(
value, backward_compat=False, **kwargs
)
Returns a log probability density together with a TangentSpace
.
A TangentSpace
allows us to calculate the correct pushforward
density when we apply a transformation to a Distribution
on
a strict submanifold of R^n (typically via a Bijector
in the
TransformedDistribution
subclass). The density correction uses
the basis of the tangent space.
Args  

value

float or double Tensor .

backward_compat

bool specifying whether to fall back to returning
FullSpace as the tangent space, and representing R^n with the standard
basis.

**kwargs

Named arguments forwarded to subclass implementation. 
Returns  

log_prob

a Tensor representing the log probability density, of shape
sample_shape(x) + self.batch_shape with values of type self.dtype .

tangent_space

a TangentSpace object (by default FullSpace )
representing the tangent space to the manifold at value .

Raises  

UnspecifiedTangentSpaceError if backward_compat is False and
the _experimental_tangent_space attribute has not been defined.

experimental_sample_and_log_prob
experimental_sample_and_log_prob(
sample_shape=(), seed=None, name='sample_and_log_prob', **kwargs
)
Samples from this distribution and returns the log density of the sample.
The default implementation simply calls sample
and log_prob
:
def _sample_and_log_prob(self, sample_shape, seed, **kwargs):
x = self.sample(sample_shape=sample_shape, seed=seed, **kwargs)
return x, self.log_prob(x, **kwargs)
However, some subclasses may provide more efficient and/or numerically stable implementations.
Args  

sample_shape

integer Tensor desired shape of samples to draw.
Default value: () .

seed

PRNG seed; see tfp.random.sanitize_seed for details.
Default value: None .

name

name to give to the op.
Default value: 'sample_and_log_prob' .

**kwargs

Named arguments forwarded to subclass implementation. 
Returns  

samples

a Tensor , or structure of Tensor s, with prepended dimensions
sample_shape .

log_prob

a Tensor of shape sample_shape(x) + self.batch_shape with
values of type self.dtype .

is_scalar_batch
is_scalar_batch(
name='is_scalar_batch'
)
Indicates that batch_shape == []
.
Args  

name

Python str prepended to names of ops created by this function.

Returns  

is_scalar_batch

bool scalar Tensor .

is_scalar_event
is_scalar_event(
name='is_scalar_event'
)
Indicates that event_shape == []
.
Args  

name

Python str prepended to names of ops created by this function.

Returns  

is_scalar_event

bool scalar Tensor .

kl_divergence
kl_divergence(
other, name='kl_divergence'
)
Computes the KullbackLeibler divergence.
Denote this distribution (self
) by p
and the other
distribution by
q
. Assuming p, q
are absolutely continuous with respect to reference
measure r
, the KL divergence is defined as:
KL[p, q] = E_p[log(p(X)/q(X))]
= int_F p(x) log q(x) dr(x) + int_F p(x) log p(x) dr(x)
= H[p, q]  H[p]
where F
denotes the support of the random variable X ~ p
, H[., .]
denotes (Shannon) cross entropy, and H[.]
denotes (Shannon) entropy.
Args  

other

tfp.distributions.Distribution instance.

name

Python str prepended to names of ops created by this function.

Returns  

kl_divergence

self.dtype Tensor with shape [B1, ..., Bn]
representing n different calculations of the KullbackLeibler
divergence.

l_ensemble_matrix
l_ensemble_matrix()
Returns the Lensemble parameterization of the DPP.
log_cdf
log_cdf(
value, name='log_cdf', **kwargs
)
Log cumulative distribution function.
Given random variable X
, the cumulative distribution function cdf
is:
log_cdf(x) := Log[ P[X <= x] ]
Often, a numerical approximation can be used for log_cdf(x)
that yields
a more accurate answer than simply taking the logarithm of the cdf
when
x << 1
.
Args  

value

float or double Tensor .

name

Python str prepended to names of ops created by this function.

**kwargs

Named arguments forwarded to subclass implementation. 
Returns  

logcdf

a Tensor of shape sample_shape(x) + self.batch_shape with
values of type self.dtype .

log_prob
log_prob(
value, name='log_prob', **kwargs
)
Log probability density/mass function.
Args  

value

float or double Tensor .

name

Python str prepended to names of ops created by this function.

**kwargs

Named arguments forwarded to subclass implementation. 
Returns  

log_prob

a Tensor of shape sample_shape(x) + self.batch_shape with
values of type self.dtype .

log_survival_function
log_survival_function(
value, name='log_survival_function', **kwargs
)
Log survival function.
Given random variable X
, the survival function is defined:
log_survival_function(x) = Log[ P[X > x] ]
= Log[ 1  P[X <= x] ]
= Log[ 1  cdf(x) ]
Typically, different numerical approximations can be used for the log
survival function, which are more accurate than 1  cdf(x)
when x >> 1
.
Args  

value

float or double Tensor .

name

Python str prepended to names of ops created by this function.

**kwargs

Named arguments forwarded to subclass implementation. 
Returns  

Tensor of shape sample_shape(x) + self.batch_shape with values of type
self.dtype .

marginal_kernel
marginal_kernel()
Returns the marginal kernel that defines the DPP.
marginal_log_prob
marginal_log_prob(
value
)
Computes the marginal log probability of an event.
The marginal log probability is the logprobability that a set sampled from
the DPP will include value
as a subset. By contrast, log_prob
returns
the logprobability of sampling exactly value
.
Args  

value

Tensor broadcastable to [batch_shape, n_points] corresponding
to the onehot encoding of a subset of points.

Returns  

The log marginal probability of value according to the DPP.

mean
mean(
name='mean', **kwargs
)
Mean.
mode
mode(
name='mode', **kwargs
)
Mode.
param_shapes
@classmethod
param_shapes( sample_shape, name='DistributionParamShapes' )
Shapes of parameters given the desired shape of a call to sample()
.
This is a class method that describes what key/value arguments are required
to instantiate the given Distribution
so that a particular shape is
returned for that instance's call to sample()
.
Subclasses should override class method _param_shapes
.
Args  

sample_shape

Tensor or python list/tuple. Desired shape of a call to
sample() .

name

name to prepend ops with. 
Returns  

dict of parameter name to Tensor shapes.

param_static_shapes
@classmethod
param_static_shapes( sample_shape )
param_shapes with static (i.e. TensorShape
) shapes.
This is a class method that describes what key/value arguments are required
to instantiate the given Distribution
so that a particular shape is
returned for that instance's call to sample()
. Assumes that the sample's
shape is known statically.
Subclasses should override class method _param_shapes
to return
constantvalued tensors when constant values are fed.
Args  

sample_shape

TensorShape or python list/tuple. Desired shape of a call
to sample() .

Returns  

dict of parameter name to TensorShape .

Raises  

ValueError

if sample_shape is a TensorShape and is not fully defined.

parameter_properties
@classmethod
parameter_properties( dtype=tf.float32, num_classes=None )
Returns a dict mapping constructor arg names to property annotations.
This dict should include an entry for each of the distribution's
Tensor
valued constructor arguments.
Distribution subclasses are not required to implement
_parameter_properties
, so this method may raise NotImplementedError
.
Providing a _parameter_properties
implementation enables several advanced
features, including:
 Distribution batch slicing (
sliced_distribution = distribution[i:j]
).  Automatic inference of
_batch_shape
and_batch_shape_tensor
, which must otherwise be computed explicitly.  Automatic instantiation of the distribution within TFP's internal property tests.
 Automatic construction of 'trainable' instances of the distribution using appropriate bijectors to avoid violating parameter constraints. This enables the distribution family to be used easily as a surrogate posterior in variational inference.
In the future, parameter property annotations may enable additional
functionality; for example, returning Distribution instances from
tf.vectorized_map
.
Args  

dtype

Optional float dtype to assume for continuousvalued parameters.
Some constraining bijectors require advance knowledge of the dtype
because certain constants (e.g., tfb.Softplus.low ) must be
instantiated with the same dtype as the values to be transformed.

num_classes

Optional int Tensor number of classes to assume when
inferring the shape of parameters for categoricallike distributions.
Otherwise ignored.

Returns  

parameter_properties

A
str > tfp.python.internal.parameter_properties.ParameterPropertiesdict mapping constructor argument names to ParameterProperties`
instances.

Raises  

NotImplementedError

if the distribution class does not implement
_parameter_properties .

prob
prob(
value, name='prob', **kwargs
)
Probability density/mass function.
Args  

value

float or double Tensor .

name

Python str prepended to names of ops created by this function.

**kwargs

Named arguments forwarded to subclass implementation. 
Returns  

prob

a Tensor of shape sample_shape(x) + self.batch_shape with
values of type self.dtype .

quantile
quantile(
value, name='quantile', **kwargs
)
Quantile function. Aka 'inverse cdf' or 'percent point function'.
Given random variable X
and p in [0, 1]
, the quantile
is:
quantile(p) := x such that P[X <= x] == p
Args  

value

float or double Tensor .

name

Python str prepended to names of ops created by this function.

**kwargs

Named arguments forwarded to subclass implementation. 
Returns  

quantile

a Tensor of shape sample_shape(x) + self.batch_shape with
values of type self.dtype .

sample
sample(
sample_shape=(), seed=None, name='sample', **kwargs
)
Generate samples of the specified shape.
Note that a call to sample()
without arguments will generate a single
sample.
Args  

sample_shape

0D or 1D int32 Tensor . Shape of the generated samples.

seed

PRNG seed; see tfp.random.sanitize_seed for details.

name

name to give to the op. 
**kwargs

Named arguments forwarded to subclass implementation. 
Returns  

samples

a Tensor with prepended dimensions sample_shape .

stddev
stddev(
name='stddev', **kwargs
)
Standard deviation.
Standard deviation is defined as,
stddev = E[(X  E[X])**2]**0.5
where X
is the random variable associated with this distribution, E
denotes expectation, and stddev.shape = batch_shape + event_shape
.
Args  

name

Python str prepended to names of ops created by this function.

**kwargs

Named arguments forwarded to subclass implementation. 
Returns  

stddev

Floatingpoint Tensor with shape identical to
batch_shape + event_shape , i.e., the same shape as self.mean() .

survival_function
survival_function(
value, name='survival_function', **kwargs
)
Survival function.
Given random variable X
, the survival function is defined:
survival_function(x) = P[X > x]
= 1  P[X <= x]
= 1  cdf(x).
Args  

value

float or double Tensor .

name

Python str prepended to names of ops created by this function.

**kwargs

Named arguments forwarded to subclass implementation. 
Returns  

Tensor of shape sample_shape(x) + self.batch_shape with values of type
self.dtype .

unnormalized_log_prob
unnormalized_log_prob(
value, name='unnormalized_log_prob', **kwargs
)
Potentially unnormalized log probability density/mass function.
This function is similar to log_prob
, but does not require that the
return value be normalized. (Normalization here refers to the total
integral of probability being one, as it should be by definition for any
probability distribution.) This is useful, for example, for distributions
where the normalization constant is difficult or expensive to compute. By
default, this simply calls log_prob
.
Args  

value

float or double Tensor .

name

Python str prepended to names of ops created by this function.

**kwargs

Named arguments forwarded to subclass implementation. 
Returns  

unnormalized_log_prob

a Tensor of shape
sample_shape(x) + self.batch_shape with values of type self.dtype .

variance
variance(
name='variance', **kwargs
)
Variance.
Variance is defined as,
Var = E[(X  E[X])**2]
where X
is the random variable associated with this distribution, E
denotes expectation, and Var.shape = batch_shape + event_shape
.
Args  

name

Python str prepended to names of ops created by this function.

**kwargs

Named arguments forwarded to subclass implementation. 
Returns  

variance

Floatingpoint Tensor with shape identical to
batch_shape + event_shape , i.e., the same shape as self.mean() .

__getitem__
__getitem__(
slices
)
Slices the batch axes of this distribution, returning a new instance.
b = tfd.Bernoulli(logits=tf.zeros([3, 5, 7, 9]))
b.batch_shape # => [3, 5, 7, 9]
b2 = b[:, tf.newaxis, ..., 2:, 1::2]
b2.batch_shape # => [3, 1, 5, 2, 4]
x = tf.random.stateless_normal([5, 3, 2, 2])
cov = tf.matmul(x, x, transpose_b=True)
chol = tf.linalg.cholesky(cov)
loc = tf.random.stateless_normal([4, 1, 3, 1])
mvn = tfd.MultivariateNormalTriL(loc, chol)
mvn.batch_shape # => [4, 5, 3]
mvn.event_shape # => [2]
mvn2 = mvn[:, 3:, ..., ::1, tf.newaxis]
mvn2.batch_shape # => [4, 2, 3, 1]
mvn2.event_shape # => [2]
Args  

slices

slices from the [] operator 
Returns  

dist

A new tfd.Distribution instance with sliced parameters.

__iter__
__iter__()