tfp.distributions.DeterminantalPointProcess

Determinantal point process (DPP) distribution.

Inherits From: Distribution

The DPP disribution parameterized by the eigenvalues and eigenvectors of the L-ensemble matrix. The L-ensemble 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 L-ensemble matrix. It assigns to any subset S of {1, ..., n} the probability:

Pr(S) = det(L_S) / det(I + L)

where:

  • L is the L-ensemble matrix parameterized by eigenvalues and eigenvectors, i.e. L = U D U^T for U = eigenvectors and D = eigenvalues.
  • L_S is the principal submatrix of L indexed by items in S. 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 s1th 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 (E-DPP) is built by sampling a subset of eigenvectors S from a Bernoulli distribution with probs equal to D / (D + 1). This E-DPP has the same eigenvectors U as L, but its eigenvalues are 1 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 point i, the kernel is updated by projecting it onto the subspace of eigenvectors orthogonal to the ith basis vector.

Examples

Sample points on the unit square grid:

import itertools
import tensorflow as tf
import tensorflow_probability 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 inner-most dimension of the result of `dpp.sample` is a multi-hot
# 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

eigenvalues float Tensor representing the eigenvalues of the DPP kernel (a.k.a. "L"). All eigenvalues must be > 0. Shape has the form [b1, ..., bB, n] where n is the number of points in the ground set.
eigenvectors float Tensor representing the column eigenvectors of the DPP kernel ("L"), provided in the same order as the eigenvalues. Shape has the form [b1, ..., bB, n, n] where n is the number of points in the ground set. The batch shape components need not be identical to those of eigenvalues, but must be broadcast compatible with them.
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.
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. Default value: False.
name Python str name prefixed to ops created by this class.

allow_nan_stats Python bool describing behavior when a stat is undefined.

Stats return +/- infinity when it makes sense. E.g., the variance of a Cauchy distribution is infinity. However, sometimes the statistic is undefined, e.g., if a distribution's pdf does not achieve a maximum within the support of the distribution, the mode is undefined. If the mean is undefined, then by definition the variance is undefined. E.g. the mean for Student's T for df = 1 is undefined (no clear way to say it is either + or - infinity), so the variance = E[(X - mean)**2] is also undefined.

batch_shape Shape of a single sample from a single event index as a TensorShape.

May be partially defined or unknown.

The batch dimensions are indexes into independent, non-identical parameterizations of this distribution.

dtype The DType of Tensors handled by this Distribution.
eigenvalues

eigenvectors

event_shape Shape of a single sample from a single batch as a TensorShape.

May be partially defined or unknown.

name Name prepended to all ops created by this Distribution.
name_scope Returns a tf.name_scope instance for this class.
non_trainable_variables Sequence of non-trainable variables owned by this module and its submodules.

parameters Dictionary of parameters used to instantiate this Distribution.
reparameterization_type Describes how samples from the distribution are reparameterized.

Currently this is one of the static instances tfd.FULLY_REPARAMETERIZED or tfd.NOT_REPARAMETERIZED.

submodules Sequence of all sub-modules.

Submodules are modules which are properties of this module, or found as properties of modules which are properties of this module (and so on).

a = tf.Module()
b = tf.Module()
c = tf.Module()
a.b = b
b.c = c
list(a.submodules) == [b, c]
True
list(b.submodules) == [c]
True
list(c.submodules) == []
True

trainable_variables Sequence of trainable variables owned by this module and its submodules.

validate_args Python bool indicating possibly expensive checks are enabled.
variables Sequence of variables owned by this module and its submodules.

Methods

batch_shape_tensor

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Shape of a single sample from a single event index as a 1-D Tensor.

The batch dimensions are indexes into independent, non-identical parameterizations of this distribution.

Args
name name to give to the op

Returns
batch_shape Tensor.

cdf

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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

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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

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Covariance.

Covariance is (possibly) defined only for non-scalar-event distributions.

For example, for a length-k, vector-valued 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 non-vector, multivariate distributions (e.g., matrix-valued, 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 length-k' vector.

Args
name Python str prepended to names of ops created by this function.
**kwargs Named arguments forwarded to subclass implementation.

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

cross_entropy

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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

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Shannon entropy in nats.

event_shape_tensor

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Shape of a single sample from a single batch as a 1-D int32 Tensor.

Args
name name to give to the op

Returns
event_shape Tensor.

experimental_default_event_space_bijector

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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 * (k-1) // 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.

is_scalar_batch

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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

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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

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Computes the Kullback--Leibler 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 Kullback-Leibler divergence.

l_ensemble_matrix

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Returns the L-ensemble parameterization of the DPP.

log_cdf

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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

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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

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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

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Returns the marginal kernel that defines the DPP.

marginal_log_prob

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Computes the marginal log probability of an event.

The marginal log probability is the log-probability that a set sampled from the DPP will include value as a subset. By contrast, log_prob returns the log-probability of sampling exactly value.

Args
value Tensor broadcastable to [batch_shape, n_points] corresponding to the one-hot encoding of a subset of points.

Returns
The log marginal probability of value according to the DPP.

mean

View source

Mean.

mode

View source

Mode.

param_shapes

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Shapes of parameters given the desired shape of a call to sample(). (deprecated)

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

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param_shapes with static (i.e. TensorShape) shapes. (deprecated)

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 constant-valued 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

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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.

Args
dtype Optional float dtype to assume for continuous-valued 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 categorical-like distributions. Otherwise ignored.

Returns
parameter_properties A str ->tfp.python.internal.parameter_properties.ParameterPropertiesdict mapping constructor argument names toParameterProperties` instances.

prob

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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

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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

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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 Python integer or tfp.util.SeedStream instance, for seeding PRNG.
name name to give to the op.
**kwargs Named arguments forwarded to subclass implementation.

Returns
samples a Tensor with prepended dimensions sample_shape.

stddev

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

survival_function

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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.

variance

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

with_name_scope

Decorator to automatically enter the module name scope.

class MyModule(tf.Module):
  @tf.Module.with_name_scope
  def __call__(self, x):
    if not hasattr(self, 'w'):
      self.w = tf.Variable(tf.random.normal([x.shape[1], 3]))
    return tf.matmul(x, self.w)

Using the above module would produce tf.Variables and tf.Tensors whose names included the module name:

mod = MyModule()
mod(tf.ones([1, 2]))
<tf.Tensor: shape=(1, 3), dtype=float32, numpy=..., dtype=float32)>
mod.w
<tf.Variable 'my_module/Variable:0' shape=(2, 3) dtype=float32,
numpy=..., dtype=float32)>

Args
method The method to wrap.

Returns
The original method wrapped such that it enters the module's name scope.

__getitem__

View source

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.normal([5, 3, 2, 2])
cov = tf.matmul(x, x, transpose_b=True)
chol = tf.linalg.cholesky(cov)
loc = tf.random.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__

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