tfp.distributions.PowerSpherical

The Power Spherical distribution over unit vectors on S^{n-1}.

Inherits From: Distribution

The Power Spherical distribution is a distribution over vectors on the unit hypersphere S^{n-1} embedded in n dimensions (R^n).

It serves as an alternative to the von Mises-Fisher distribution with a simpler (faster) log_prob calculation, as well as a reparameterizable sampler. In contrast, the Power Spherical distribution does have -mean_direction as a point with zero density (and hence a neighborhood around that having arbitrarily small density), in contrast with the von Mises-Fisher distribution which has non-zero density everywhere.

Mathematical details

The probability density function (pdf) is,

pdf(x; mu, kappa) = C(kappa) (1 + mu^T x) ** k
where,
C(kappa) = 2**(a + b) pi**b Gamma(a) / Gamma(a + b)
a = (n - 1) / 2. + k
b = (n - 1) / 2.

where:

  • mean_direction = mu; a unit vector in R^k,
  • concentration = kappa; scalar real >= 0, concentration of samples around mean_direction, where 0 pertains to the uniform distribution on the hypersphere, and \inf indicates a delta function at mean_direction.

Examples

A single instance of a PowerSpherical distribution is defined by a mean direction unit vector.

Extra leading dimensions, if provided, allow for batches.

tfd = tfp.distributions

# Initialize a single 3-dimension PowerSpherical distribution.
mu = [0., 1, 0]
conc = 1.
ps = tfd.PowerSpherical(mean_direction=mu, concentration=conc)

# Evaluate this on an observation in S^2 (in R^3), returning a scalar.
ps.prob([1., 0, 0])

# Initialize a batch of two 3-variate vMF distributions.
mu = [[0., 1, 0],
      [1., 0, 0]]
conc = [1., 2]
ps = tfd.PowerSpherical(mean_direction=mu, concentration=conc)

# Evaluate this on two observations, each in S^2, returning a length two
# tensor.
x = [[0., 0, 1],
     [0., 1, 0]]
ps.prob(x)

#### References

[1] Nicola de Cao, Wilker Aziz. The Power Spherical distribution.
    https://arxiv.org/abs/2006.04437.

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<colgroup><col width="214px"><col></colgroup>
<tr><th colspan="2"><h2 class="add-link">Args</h2></th></tr>

<tr>
<td>
`mean_direction`
</td>
<td>
Floating-point `Tensor` with shape [B1, ... Bn, N].
A unit vector indicating the mode of the distribution, or the
unit-normalized direction of the mean.
</td>
</tr><tr>
<td>
`concentration`
</td>
<td>
Floating-point `Tensor` having batch shape [B1, ... Bn]
broadcastable with `mean_direction`. The level of concentration of
samples around the `mean_direction`. `concentration=0` indicates a
uniform distribution over the unit hypersphere, and `concentration=+inf`
indicates a `Deterministic` distribution (delta function) at
`mean_direction`.
</td>
</tr><tr>
<td>
`validate_args`
</td>
<td>
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.
</td>
</tr><tr>
<td>
`allow_nan_stats`
</td>
<td>
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.
</td>
</tr><tr>
<td>
`name`
</td>
<td>
Python `str` name prefixed to Ops created by this class.
</td>
</tr>
</table>



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<tr><th colspan="2"><h2 class="add-link">Raises</h2></th></tr>

<tr>
<td>
`ValueError`
</td>
<td>
For known-bad arguments, i.e. unsupported event dimension.
</td>
</tr>
</table>





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<tr><th colspan="2"><h2 class="add-link">Attributes</h2></th></tr>

<tr>
<td>
`allow_nan_stats`
</td>
<td>
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.
</td>
</tr><tr>
<td>
`batch_shape`
</td>
<td>
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.
</td>
</tr><tr>
<td>
`concentration`
</td>
<td>
Concentration parameter.
</td>
</tr><tr>
<td>
`dtype`
</td>
<td>
The `DType` of `Tensor`s handled by this `Distribution`.
</td>
</tr><tr>
<td>
`event_shape`
</td>
<td>
Shape of a single sample from a single batch as a `TensorShape`.

May be partially defined or unknown.
</td>
</tr><tr>
<td>
`mean_direction`
</td>
<td>
Mean direction parameter.
</td>
</tr><tr>
<td>
`name`
</td>
<td>
Name prepended to all ops created by this `Distribution`.
</td>
</tr><tr>
<td>
`name_scope`
</td>
<td>
Returns a <a href="https://www.tensorflow.org/api_docs/python/tf/name_scope"><code>tf.name_scope</code></a> instance for this class.
</td>
</tr><tr>
<td>
`parameters`
</td>
<td>
Dictionary of parameters used to instantiate this `Distribution`.
</td>
</tr><tr>
<td>
`reparameterization_type`
</td>
<td>
Describes how samples from the distribution are reparameterized.

Currently this is one of the static instances
`tfd.FULLY_REPARAMETERIZED` or `tfd.NOT_REPARAMETERIZED`.
</td>
</tr><tr>
<td>
`submodules`
</td>
<td>
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).

<pre class="devsite-click-to-copy prettyprint lang-py">
<code class="devsite-terminal" data-terminal-prefix="&gt;&gt;&gt;">a = tf.Module()</code>
<code class="devsite-terminal" data-terminal-prefix="&gt;&gt;&gt;">b = tf.Module()</code>
<code class="devsite-terminal" data-terminal-prefix="&gt;&gt;&gt;">c = tf.Module()</code>
<code class="devsite-terminal" data-terminal-prefix="&gt;&gt;&gt;">a.b = b</code>
<code class="devsite-terminal" data-terminal-prefix="&gt;&gt;&gt;">b.c = c</code>
<code class="devsite-terminal" data-terminal-prefix="&gt;&gt;&gt;">list(a.submodules) == [b, c]</code>
<code class="no-select nocode">True</code>
<code class="devsite-terminal" data-terminal-prefix="&gt;&gt;&gt;">list(b.submodules) == [c]</code>
<code class="no-select nocode">True</code>
<code class="devsite-terminal" data-terminal-prefix="&gt;&gt;&gt;">list(c.submodules) == []</code>
<code class="no-select nocode">True</code>
</pre>

</td>
</tr><tr>
<td>
`trainable_variables`
</td>
<td>
Sequence of trainable variables owned by this module and its submodules.

Note: this method uses reflection to find variables on the current instance
and submodules. For performance reasons you may wish to cache the result
of calling this method if you don't expect the return value to change.
</td>
</tr><tr>
<td>
`validate_args`
</td>
<td>
Python `bool` indicating possibly expensive checks are enabled.
</td>
</tr><tr>
<td>
`variables`
</td>
<td>
Sequence of variables owned by this module and its submodules.

Note: this method uses reflection to find variables on the current instance
and submodules. For performance reasons you may wish to cache the result
of calling this method if you don't expect the return value to change.
</td>
</tr>
</table>



## Methods

<h3 id="batch_shape_tensor"><code>batch_shape_tensor</code></h3>

<a target="_blank" href="https://github.com/tensorflow/probability/blob/v0.11.1/tensorflow_probability/python/distributions/distribution.py#L772-L805">View source</a>

<pre class="devsite-click-to-copy prettyprint lang-py tfo-signature-link">
<code>batch_shape_tensor(
    name='batch_shape_tensor'
)
</code></pre>

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.

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<tr><th colspan="2">Args</th></tr>

<tr>
<td>
`name`
</td>
<td>
name to give to the op
</td>
</tr>
</table>



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<tr><th colspan="2">Returns</th></tr>

<tr>
<td>
`batch_shape`
</td>
<td>
`Tensor`.
</td>
</tr>
</table>



<h3 id="cdf"><code>cdf</code></h3>

<a target="_blank" href="https://github.com/tensorflow/probability/blob/v0.11.1/tensorflow_probability/python/distributions/distribution.py#L1037-L1055">View source</a>

<pre class="devsite-click-to-copy prettyprint lang-py tfo-signature-link">
<code>cdf(
    value, name='cdf', **kwargs
)
</code></pre>

Cumulative distribution function.

Given random variable `X`, the cumulative distribution function `cdf` is:

```none
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

View source

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

View source

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

View source

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.

other types with built-in registrations: SphericalUniform, VonMisesFisher

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

View source

Shannon entropy in nats.

event_shape_tensor

View source

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.

is_scalar_batch

View source

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

View source

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

View source

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.

other types with built-in registrations: SphericalUniform, VonMisesFisher

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.

log_cdf

View source

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

View source

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

View source

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.

mean

View source

Mean.

mode

View source

Mode.

param_shapes

View source

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

View source

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 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.

prob

View source

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

View source

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

View source

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

View source

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

View source

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

View source

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.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__

View source