positive_semidefinite_kernel.py

# Copyright 2018 The TensorFlow Probability Authors.
#
# Licensed under the Apache License, Version 2.0 (the "License");
# you may not use this file except in compliance with the License.
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#     http://www.apache.org/licenses/LICENSE-2.0
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# ============================================================================
"""PositiveSemidefiniteKernel base."""

from __future__ import absolute_import
from __future__ import division
from __future__ import print_function

import abc
import contextlib
import functools
import operator
import six
import tensorflow.compat.v2 as tf

from tensorflow_probability.python.math.psd_kernels.internal import util


__all__ = [
    'PositiveSemidefiniteKernel',
]


@six.add_metaclass(abc.ABCMeta)
class PositiveSemidefiniteKernel(tf.Module):
  """Abstract base class for positive semi-definite kernel functions.

  #### Background

  For any set `S`, a real- (or complex-valued) function `k` on the Cartesian
  product `S x S` is called positive semi-definite if we have

  ```none
  sum_i sum_j (c[i]*) c[j] k(x[i], x[j]) >= 0
  ```

  for any finite collections `{x[1], ..., x[N]}` in S and `{c[1], ..., c[N]}` in
  the reals (or the complex plane). '*' denotes the complex conjugate, in the
  complex case.

  Some examples:
    - `S` is R, and `k(s, t) = (s - a) (t - b)`, where a, b are in R. This
      corresponds to a linear kernel.
    - `S` is R^+ U {0}, and `k(s, t) = min(s, t)`. This corresponds to a kernel
      for a Wiener process.
    - `S` is the set of strings over an alphabet `A = {c1, ... cC}`, and
      `k(s, t)` is defined via some similarity metric over strings.

  We model positive semi-definite functions (*kernels*, in common machine
  learning parlance) as classes with 3 primary public methods: `apply`,
  `matrix`, and `tensor`.

  `apply` computes the value of the kernel function at a pair of (batches of)
  input locations. It is the more "low-level" operation: `matrix` and `tensor`
  are implemented in terms of `apply`.

  `matrix` computes the value of the kernel *pairwise* on two (batches of)
  lists of input examples. When the two collections are the same the result is
  called the Gram (or Gramian) matrix
  (https://en.wikipedia.org/wiki/Gramian_matrix).

  `tensor` generalizes `matrix`, taking rank `k1` and `k2` collections of
  input examples to a rank `k1 + k2` collection of kernel values.

  #### Kernel Parameter Shape Semantics

  PositiveSemidefiniteKernel implementations support batching of kernel
  parameters and broadcasting of these parameters across batches of inputs. This
  allows, for example, creating a single kernel object which acts like a
  collection of kernels with different parameters. This might be useful for,
  e.g., for exploring multiple random initializations in parallel during a
  kernel parameter optimization procedure.

  The interaction between kernel parameter shapes and input shapes (see below)
  is somewhat subtle. The semantics are designed to make the most common use
  cases easy, while not ruling out more intricate control. The overarching
  principle is that kernel parameter batch shapes must be broadcastable with
  input batch shapes (see below). Examples are provided in the method-level
  documentation.

  #### Input Shape Semantics

  PositiveSemidefiniteKernel methods each support a notion of batching inputs;
  see the method-level documentation for full details; here we describe the
  overall semantics of input shapes. Inputs to PositiveSemidefiniteKernel
  methods partition into 3 pieces:

  ```none
  [b1, ..., bB, e1, ..., eE, f1, ..., fF]
  '----------'  '---------'  '---------'
       |             |            '-- Feature dimensions
       |             '-- Example dimensions
       '-- Batch dimensions
  ```

  - Feature dimensions correspond to the space over which the kernel is defined;
    in typical applications inputs are vectors and this part of the shape is
    rank-1. For example, if our kernel is defined over R^2 x R^2, each input is
    a 2-D vector (a rank-1 tensor of shape `[2,]`) so that
    `F = 1, [f1, ..., fF] = [2]`. If we defined a kernel over DxD matrices, its
    domain would be R^(DxD) x R^(DxD), we would have `F = 2` and
    `[f1, ..., fF] = [D, D]`. Feature shapes of inputs should be the same, but
    no exception will be raised unless they are broadcast-incompatible.
  - Batch dimensions describe collections of inputs which in some sense have
    nothing to do with each other, but may be coupled to batches of kernel
    parameters. It's required that batch dimensions of inputs broadcast with
    each other, and with the kernel's overall batch shape.
  - Example dimensions are shape elements which represent a collection of inputs
    that in some sense "go together" (whereas batches are "independent"). The
    exact semantics are different for the `apply`, `matrix` and `tensor` methods
    (see method-level doc strings for more details). `apply` combines examples
    together pairwise, much like the python built-in `zip`. `matrix` combines
    examples pairwise for *all* pairs of elements from two rank-1 input
    collections (lists), ie, it applies the kernel to all elements in the
    cross-product of two lists of examples. `tensor` further generalizes
    `matrix` to higher rank collections of inputs. Only `matrix` strictly
    requires example dimensions to be present (and to be exactly rank 1),
    although the typical usage of `apply` (eg, building a matrix diagonal) will
    also have `example_ndims` 1.

  ##### Examples

    ```python
    import tensorflow_probability as tfp

    # Suppose `SomeKernel` acts on vectors (rank-1 tensors), ie number of
    # feature dimensions is 1.
    scalar_kernel = tfp.math.psd_kernels.SomeKernel(param=.5)
    scalar_kernel.batch_shape
    # ==> []

    # `x` and `y` are batches of five 3-D vectors:
    x = np.ones([5, 3], np.float32)
    y = np.ones([5, 3], np.float32)
    scalar_kernel.apply(x, y).shape
    # ==> [5]

    scalar_kernel.matrix(x, y).shape
    # ==> [5, 5]
    ```

    Now we can consider a kernel with batched parameters:

    ```python
    batch_kernel = tfp.math.psd_kernels.SomeKernel(param=[.2, .5])
    batch_kernel.batch_shape
    # ==> [2]

    # `x` and `y` are batches of five 3-D vectors:
    x = np.ones([5, 3], np.float32)
    y = np.ones([5, 3], np.float32)

    batch_kernel.apply(x, y).shape
    # ==> Error! [2] and [5] can't broadcast.
    # We could solve this by telling `apply` to treat the 5 as an example dim:

    batch_kernel.apply(x, y, example_ndims=1).shape
    # ==> [2, 5]

    # Note that example_ndims is implicitly 1 for a call to `matrix`, so the
    # following just works:
    batch_kernel.matrix(x, y).shape
    # ==> [2, 5, 5]
    ```

  """

  def __init__(self, feature_ndims, dtype=None, name=None, validate_args=False):
    """Construct a PositiveSemidefiniteKernel (subclass) instance.

    Args:
      feature_ndims: Python `integer` indicating the number of dims (the rank)
        of the feature space this kernel acts on.
      dtype: `DType` on which this kernel operates.
      name: Python `str` name prefixed to Ops created by this class. Default:
        subclass name.
      validate_args: Python `bool`, default `False`. When `True` kernel
        parameters are checked for validity despite possibly degrading runtime
        performance. When `False` invalid inputs may silently render incorrect
        outputs.

    Raises:
      ValueError: if `feature_ndims` is not an integer greater than 0
    Inputs to PositiveSemidefiniteKernel methods partition into 3 pieces:

    ```none
    [b1, ..., bB, e1, ..., eE, f1, ..., fF]
    '----------'  '---------'  '---------'
         |             |            '-- Feature dimensions
         |             '-- Example dimensions
         '-- Batch dimensions
    ```

    The `feature_ndims` argument declares how many of the right-most shape
    dimensions belong to the feature dimensions. This enables us to predict
    which shape dimensions will be 'reduced' away during kernel computation.
    """
    if not (isinstance(feature_ndims, int) and feature_ndims > 0):
      raise ValueError(
          '`feature_ndims` must be a Python `integer` greater than zero. ' +
          'Got: {}'.format(feature_ndims))
    self._feature_ndims = feature_ndims
    self._dtype = dtype
    if not name or name[-1] != '/':  # `name` is not a name scope
      name = tf.name_scope(name or type(self).__name__).name
    self._name = name
    self._validate_args = validate_args
    self._initial_parameter_control_dependencies = tuple(
        d for d in self._parameter_control_dependencies(is_init=True)
        if d is not None)
    if self._initial_parameter_control_dependencies:
      self._initial_parameter_control_dependencies = (
          tf.group(*self._initial_parameter_control_dependencies),)

  @property
  def feature_ndims(self):
    """The number of feature dimensions.

    Kernel functions generally act on pairs of inputs from some space like

    ```none
    R^(d1 x ... x dD)
    ```

    or, in words: rank-`D` real-valued tensors of shape `[d1, ..., dD]`. Inputs
    can be vectors in some `R^N`, but are not restricted to be. Indeed, one
    might consider kernels over matrices, tensors, or even more general spaces,
    like strings or graphs.

    Returns:
      The number of feature dimensions (feature rank) of this kernel.
    """
    return self._feature_ndims

  @property
  def dtype(self):
    """DType over which the kernel operates."""
    return self._dtype

  @property
  def name(self):
    """Name prepended to all ops created by this class."""
    return self._name

  @property
  def validate_args(self):
    """Python `bool` indicating possibly expensive checks are enabled."""
    return self._validate_args

  @property
  def batch_shape(self):
    """The batch_shape property of a PositiveSemidefiniteKernel.

    This property describes the fully broadcast shape of all kernel parameters.
    For example, consider an ExponentiatedQuadratic kernel, which is
    parameterized by an amplitude and length_scale:

    ```none
    exp_quad(x, x') := amplitude * exp(||x - x'||**2 / length_scale**2)
    ```

    The batch_shape of such a kernel is derived from broadcasting the shapes of
    `amplitude` and `length_scale`. E.g., if their shapes were

    ```python
    amplitude.shape = [2, 1, 1]
    length_scale.shape = [1, 4, 3]
    ```

    then `exp_quad`'s batch_shape would be `[2, 4, 3]`.

    Note that this property defers to the private _batch_shape method, which
    concrete implementation sub-classes are obliged to provide.

    Returns:
        `TensorShape` instance describing the fully broadcast shape of all
        kernel parameters.
    """
    return self._batch_shape()

  def batch_shape_tensor(self):
    """The batch_shape property of a PositiveSemidefiniteKernel as a `Tensor`.

    Returns:
        `Tensor` which evaluates to a vector of integers which are the
        fully-broadcast shapes of the kernel parameters.
    """
    with tf.name_scope(self._name):
      if self.batch_shape.is_fully_defined():
        return tf.convert_to_tensor(
            self.batch_shape.as_list(), dtype=tf.int32, name='batch_shape')
      with tf.name_scope('batch_shape_tensor'):
        return self._batch_shape_tensor()

  @contextlib.contextmanager
  def _name_and_control_scope(self, name=None):
    """Helper function to standardize op scope."""
    with tf.name_scope(self.name):
      with tf.name_scope(name) as name_scope:
        deps = tuple(
            d for d in (  # pylint: disable=g-complex-comprehension
                tuple(self._initial_parameter_control_dependencies) +
                tuple(self._parameter_control_dependencies(is_init=False))))
        if not deps:
          yield name_scope
          return
        with tf.control_dependencies(deps) as deps_scope:
          yield deps_scope

  def apply(self, x1, x2, example_ndims=0):
    """Apply the kernel function pairs of inputs.

    Args:
      x1: `Tensor` input to the kernel, of shape `B1 + E1 + F`, where `B1` and
        `E1` may be empty (ie, no batch/example dims, resp.) and `F` (the
        feature shape) must have rank equal to the kernel's `feature_ndims`
        property. Batch shape must broadcast with the batch shape of `x2` and
        with the kernel's batch shape. Example shape must broadcast with example
        shape of `x2`. `x1` and `x2` must have the same *number* of example dims
        (ie, same rank).
      x2: `Tensor` input to the kernel, of shape `B2 + E2 + F`, where `B2` and
        `E2` may be empty (ie, no batch/example dims, resp.) and `F` (the
        feature shape) must have rank equal to the kernel's `feature_ndims`
        property. Batch shape must broadcast with the batch shape of `x2` and
        with the kernel's batch shape. Example shape must broadcast with example
        shape of `x2`. `x1` and `x2` must have the same *number* of example
      example_ndims: A python integer, the number of example dims in the inputs.
        In essence, this parameter controls how broadcasting of the kernel's
        batch shape with input batch shapes works. The kernel batch shape will
        be broadcast against everything to the left of the combined example and
        feature dimensions in the input shapes.

    Returns:
      `Tensor` containing the results of applying the kernel function to inputs
      `x1` and `x2`. If the kernel parameters' batch shape is `Bk` then the
      shape of the `Tensor` resulting from this method call is
      `broadcast(Bk, B1, B2) + broadcast(E1, E2)`.

    Given an index set `S`, a kernel function is mathematically defined as a
    real- or complex-valued function on `S` satisfying the
    positive semi-definiteness constraint:

    ```none
    sum_i sum_j (c[i]*) c[j] k(x[i], x[j]) >= 0
    ```

    for any finite collections `{x[1], ..., x[N]}` in `S` and
    `{c[1], ..., c[N]}` in the reals (or the complex plane). '*' is the complex
    conjugate, in the complex case.

    This method most closely resembles the function described in the
    mathematical definition of a kernel. Given a PositiveSemidefiniteKernel `k`
    with scalar parameters and inputs `x` and `y` in `S`, `apply(x, y)` yields a
    single scalar value.

    #### Examples

    ```python
    import tensorflow_probability as tfp

    # Suppose `SomeKernel` acts on vectors (rank-1 tensors)
    scalar_kernel = tfp.math.psd_kernels.SomeKernel(param=.5)
    scalar_kernel.batch_shape
    # ==> []

    # `x` and `y` are batches of five 3-D vectors:
    x = np.ones([5, 3], np.float32)
    y = np.ones([5, 3], np.float32)
    scalar_kernel.apply(x, y).shape
    # ==> [5]
    ```

    The above output is the result of vectorized computation of the five values

    ```none
    [k(x[0], y[0]), k(x[1], y[1]), ..., k(x[4], y[4])]
    ```

    Now we can consider a kernel with batched parameters:

    ```python
    batch_kernel = tfp.math.psd_kernels.SomeKernel(param=[.2, .5])
    batch_kernel.batch_shape
    # ==> [2]
    batch_kernel.apply(x, y).shape
    # ==> Error! [2] and [5] can't broadcast.
    ```

    The parameter batch shape of `[2]` and the input batch shape of `[5]` can't
    be broadcast together. We can fix this in either of two ways:

    1. Give the parameter a shape of `[2, 1]` which will correctly
    broadcast with `[5]` to yield `[2, 5]`:

    ```python
    batch_kernel = tfp.math.psd_kernels.SomeKernel(
        param=[[.2], [.5]])
    batch_kernel.batch_shape
    # ==> [2, 1]
    batch_kernel.apply(x, y).shape
    # ==> [2, 5]
    ```

    2. By specifying `example_ndims`, which tells the kernel to treat the `5`
    in the input shape as part of the "example shape", and "pushing" the
    kernel batch shape to the left:

    ```python
    batch_kernel = tfp.math.psd_kernels.SomeKernel(param=[.2, .5])
    batch_kernel.batch_shape
    # ==> [2]
    batch_kernel.apply(x, y, example_ndims=1).shape
    # ==> [2, 5]

    """
    with self._name_and_control_scope(self._name):
      x1 = tf.convert_to_tensor(x1, name='x1', dtype_hint=self.dtype)
      x2 = tf.convert_to_tensor(x2, name='x2', dtype_hint=self.dtype)

      should_expand_dims = (example_ndims == 0)

      if should_expand_dims:
        example_ndims += 1
        x1 = tf.expand_dims(x1, -(self.feature_ndims + 1))
        x2 = tf.expand_dims(x2, -(self.feature_ndims + 1))

      result = self._apply(x1, x2, example_ndims=example_ndims)

      if should_expand_dims:
        result = tf.squeeze(result, axis=-1)

      return result

  def _apply(self, x1, x2, example_ndims=1):
    """Apply the kernel function to a pair of (batches of) inputs.

    Subclasses must implement this method. It will always be called with
    example_ndims >= 1. Implementations should take care to respect
    example_ndims, by padding parameters on the right with 1's example_ndims
    times. See tests and existing subclasses for examples.

    Args:
      x1: `Tensor` input to the first positional parameter of the kernel, of
        shape `B1 + E1 + F`, where `B1` may be empty (ie, no batch dims, resp.),
        `E1` is a shape of rank at least 1, and `F` (the feature shape) must
        have rank equal to the kernel's `feature_ndims` property. Batch shape
        must broadcast with the batch shape of `x2` and with the kernel's batch
        shape. Example shape must broadcast with example shape of `x2` (They
        don't strictly need to be equal, e.g., when `apply` is called from
        `matrix`, `x1` and `x2` each have 1's in opposing positions in their
        example shapes). `x1` and `x2` must have the same *number* of example
        dims (ie, same rank).
      x2: `Tensor` input to the second positional parameter of the kernel,
        shape `B2 + E2 + F`, where `B2` may be empty (ie, no batch dims, resp.),
        `E2` is a shape of rank at least 1, and `F` (the feature shape) must
        have rank equal to the kernel's `feature_ndims` property. Batch shape
        must broadcast with the batch shape of `x1` and with the kernel's batch
        shape. Example shape must broadcast with example shape of `x1` (They
        don't strictly need to be equal, e.g., when `apply` is called from
        `matrix`, `x1` and `x2` each have 1's in opposing positions in their
        example shapes). `x1` and `x2` must have the same *number* of example
        dims (ie, same rank).
      example_ndims: A python integer greater than or equal to 1, the number of
        example dims in the inputs. In essence, this parameter controls how
        broadcasting of the kernel's batch shape with input batch shapes works.
        The kernel batch shape will be broadcast against everything to the left
        of the combined example and feature dimensions in the input shapes.

    Returns:
      `Tensor` containing the results of applying the kernel function to inputs
      `x1` and `x2`. If the kernel parameters' batch shape is `Bk` then the
      shape of the `Tensor` resulting from this method call is
      `broadcast(Bk, B1, B2) + broadcast(E1, E2)`.
    """
    raise NotImplementedError(
        'Subclasses must provide `_apply` implementation.')

  def matrix(self, x1, x2):
    """Construct (batched) matrices from (batches of) collections of inputs.

    Args:
      x1: `Tensor` input to the first positional parameter of the kernel, of
        shape `B1 + [e1] + F`, where `B1` may be empty (ie, no batch dims,
        resp.), `e1` is a single integer (ie, `x1` has example ndims exactly 1),
        and `F` (the feature shape) must have rank equal to the kernel's
        `feature_ndims` property. Batch shape must broadcast with the batch
        shape of `x2` and with the kernel's batch shape.
      x2: `Tensor` input to the second positional parameter of the kernel,
        shape `B2 + [e2] + F`, where `B2` may be empty (ie, no batch dims,
        resp.), `e2` is a single integer (ie, `x2` has example ndims exactly 1),
        and `F` (the feature shape) must have rank equal to the kernel's
        `feature_ndims` property. Batch shape must broadcast with the batch
        shape of `x1` and with the kernel's batch shape.

    Returns:
      `Tensor` containing the matrix (possibly batched) of kernel applications
      to pairs from inputs `x1` and `x2`. If the kernel parameters' batch shape
      is `Bk` then the shape of the `Tensor` resulting from this method call is
      `broadcast(Bk, B1, B2) + [e1, e2]` (note this differs from `apply`: the
      example dimensions are concatenated, whereas in `apply` the example dims
      are broadcast together).

    Given inputs `x1` and `x2` of shapes

    ```none
    [b1, ..., bB, e1, f1, ..., fF]
    ```

    and

    ```none
    [c1, ..., cC, e2, f1, ..., fF]
    ```

    This method computes the batch of `e1 x e2` matrices resulting from applying
    the kernel function to all pairs of inputs from `x1` and `x2`. The shape
    of the batch of matrices is the result of broadcasting the batch shapes of
    `x1`, `x2`, and the kernel parameters (see examples below). As such, it's
    required that these shapes all be broadcast compatible. However, the kernel
    parameter batch shapes need not broadcast against the 'example shapes' (`e1`
    and `e2` above).

    When the two inputs are the (batches of) identical collections, the
    resulting matrix is the so-called Gram (or Gramian) matrix
    (https://en.wikipedia.org/wiki/Gramian_matrix).

    #### Examples

    First, consider a kernel with a single scalar parameter.

    ```python
    import tensorflow_probability as tfp

    scalar_kernel = tfp.math.psd_kernels.SomeKernel(param=.5)
    scalar_kernel.batch_shape
    # ==> []

    # Our inputs are two lists of 3-D vectors
    x = np.ones([5, 3], np.float32)
    y = np.ones([4, 3], np.float32)
    scalar_kernel.matrix(x, y).shape
    # ==> [5, 4]
    ```

    The result comes from applying the kernel to the entries in `x` and `y`
    pairwise, across all pairs:

      ```none
      | k(x[0], y[0])    k(x[0], y[1])  ...  k(x[0], y[3]) |
      | k(x[1], y[0])    k(x[1], y[1])  ...  k(x[1], y[3]) |
      |      ...              ...                 ...      |
      | k(x[4], y[0])    k(x[4], y[1])  ...  k(x[4], y[3]) |
      ```

    Now consider a kernel with batched parameters with the same inputs

    ```python
    batch_kernel = tfp.math.psd_kernels.SomeKernel(param=[1., .5])
    batch_kernel.batch_shape
    # ==> [2]

    batch_kernel.matrix(x, y).shape
    # ==> [2, 5, 4]
    ```

    This results in a batch of 2 matrices, one computed from the kernel with
    `param = 1.` and the other with `param = .5`.

    We also support batching of the inputs. First, let's look at that with
    the scalar kernel again.

    ```python
    # Batch of 10 lists of 5 vectors of dimension 3
    x = np.ones([10, 5, 3], np.float32)

    # Batch of 10 lists of 4 vectors of dimension 3
    y = np.ones([10, 4, 3], np.float32)

    scalar_kernel.matrix(x, y).shape
    # ==> [10, 5, 4]
    ```

    The result is a batch of 10 matrices built from the batch of 10 lists of
    input vectors. These batch shapes have to be broadcastable. The following
    will *not* work:

    ```python
    x = np.ones([10, 5, 3], np.float32)
    y = np.ones([20, 4, 3], np.float32)
    scalar_kernel.matrix(x, y).shape
    # ==> Error! [10] and [20] can't broadcast.
    ```

    Now let's consider batches of inputs in conjunction with batches of kernel
    parameters. We require that the input batch shapes be broadcastable with
    the kernel parameter batch shapes, otherwise we get an error:

    ```python
    x = np.ones([10, 5, 3], np.float32)
    y = np.ones([10, 4, 3], np.float32)

    batch_kernel = tfp.math.psd_kernels.SomeKernel(params=[1., .5])
    batch_kernel.batch_shape
    # ==> [2]
    batch_kernel.matrix(x, y).shape
    # ==> Error! [2] and [10] can't broadcast.
    ```

    The fix is to make the kernel parameter shape broadcastable with `[10]` (or
    reshape the inputs to be broadcastable!):

    ```python
    x = np.ones([10, 5, 3], np.float32)
    y = np.ones([10, 4, 3], np.float32)

    batch_kernel = tfp.math.psd_kernels.SomeKernel(
        params=[[1.], [.5]])
    batch_kernel.batch_shape
    # ==> [2, 1]
    batch_kernel.matrix(x, y).shape
    # ==> [2, 10, 5, 4]

    # Or, make the inputs broadcastable:
    x = np.ones([10, 1, 5, 3], np.float32)
    y = np.ones([10, 1, 4, 3], np.float32)

    batch_kernel = tfp.math.psd_kernels.SomeKernel(
        params=[1., .5])
    batch_kernel.batch_shape
    # ==> [2]
    batch_kernel.matrix(x, y).shape
    # ==> [10, 2, 5, 4]
    ```

    Here, we have the result of applying the kernel, with 2 different
    parameters, to each of a batch of 10 pairs of input lists.

    """
    with self._name_and_control_scope(self._name):
      x1 = tf.convert_to_tensor(x1, name='x1', dtype_hint=self.dtype)
      x2 = tf.convert_to_tensor(x2, name='x2', dtype_hint=self.dtype)

      return self.tensor(x1, x2, x1_example_ndims=1, x2_example_ndims=1)

  def tensor(self, x1, x2, x1_example_ndims, x2_example_ndims):
    """Construct (batched) tensors from (batches of) collections of inputs.

    Args:
      x1: `Tensor` input to the first positional parameter of the kernel, of
        shape `B1 + E1 + F`, where `B1` and `E1` arbitrary shapes which may be
        empty (ie, no batch/example dims, resp.), and `F` (the feature shape)
        must have rank equal to the kernel's `feature_ndims` property. Batch
        shape must broadcast with the batch shape of `x2` and with the kernel's
        batch shape.
      x2: `Tensor` input to the second positional parameter of the kernel,
        shape `B2 + E2 + F`, where `B2` and `E2` arbitrary shapes which may be
        empty (ie, no batch/example dims, resp.), and `F` (the feature shape)
        must have rank equal to the kernel's `feature_ndims` property. Batch
        shape must broadcast with the batch shape of `x1` and with the kernel's
        batch shape.
      x1_example_ndims: A python integer greater than or equal to 0, the number
        of example dims in the first input. This affects both the alignment of
        batch shapes and the shape of the final output of the function.
        Everything left of the feature shape and the example dims in `x1` is
        considered "batch shape", and must broadcast as specified above.
      x2_example_ndims: A python integer greater than or equal to 0, the number
        of example dims in the second input. This affects both the alignment of
        batch shapes and the shape of the final output of the function.
        Everything left of the feature shape and the example dims in `x1` is
        considered "batch shape", and must broadcast as specified above.

    Returns:
      `Tensor` containing (possibly batched) kernel applications to pairs from
      inputs `x1` and `x2`. If the kernel parameters' batch shape is `Bk` then
      the shape of the `Tensor` resulting from this method call is
      `broadcast(Bk, B1, B2) + E1 + E2`. Note this differs from `apply`: the
      example dimensions are concatenated, whereas in `apply` the example dims
      are broadcast together. It also differs from `matrix`: the example shapes
      are arbitrary here, and the result accrues a rank equal to the sum of the
      ranks of the input example shapes.

    #### Examples

    First, consider a kernel with a single scalar parameter.

    ```python
    import tensorflow_probability as tfp

    scalar_kernel = tfp.math.psd_kernels.SomeKernel(param=.5)
    scalar_kernel.batch_shape
    # ==> []

    # Our inputs are two rank-2 collections of 3-D vectors
    x = np.ones([5, 6, 3], np.float32)
    y = np.ones([7, 8, 3], np.float32)
    scalar_kernel.tensor(x, y, x1_example_ndims=2, x2_example_ndims=2).shape
    # ==> [5, 6, 7, 8]

    # Empty example shapes work too!
    x = np.ones([3], np.float32)
    y = np.ones([5, 3], np.float32)
    scalar_kernel.tensor(x, y, x1_example_ndims=0, x2_example_ndims=1).shape
    # ==> [5]
    ```

    The result comes from applying the kernel to the entries in `x` and `y`
    pairwise, across all pairs:

      ```none
      | k(x[0], y[0])    k(x[0], y[1])  ...  k(x[0], y[3]) |
      | k(x[1], y[0])    k(x[1], y[1])  ...  k(x[1], y[3]) |
      |      ...              ...                 ...      |
      | k(x[4], y[0])    k(x[4], y[1])  ...  k(x[4], y[3]) |
      ```

    Now consider a kernel with batched parameters.

    ```python
    batch_kernel = tfp.math.psd_kernels.SomeKernel(param=[1., .5])
    batch_kernel.batch_shape
    # ==> [2]

    # Inputs are two rank-2 collections of 3-D vectors
    x = np.ones([5, 6, 3], np.float32)
    y = np.ones([7, 8, 3], np.float32)
    scalar_kernel.tensor(x, y, x1_example_ndims=2, x2_example_ndims=2).shape
    # ==> [2, 5, 6, 7, 8]
    ```

    We also support batching of the inputs. First, let's look at that with
    the scalar kernel again.

    ```python
    # Batch of 10 lists of 5x6 collections of dimension 3
    x = np.ones([10, 5, 6, 3], np.float32)

    # Batch of 10 lists of 7x8 collections of dimension 3
    y = np.ones([10, 7, 8, 3], np.float32)

    scalar_kernel.tensor(x, y, x1_example_ndims=2, x2_example_ndims=2).shape
    # ==> [10, 5, 6, 7, 8]
    ```

    The result is a batch of 10 tensors built from the batch of 10 rank-2
    collections of input vectors. The batch shapes have to be broadcastable.
    The following will *not* work:

    ```python
    x = np.ones([10, 5, 3], np.float32)
    y = np.ones([20, 4, 3], np.float32)
    scalar_kernel.tensor(x, y, x1_example_ndims=1, x2_example_ndims=1).shape
    # ==> Error! [10] and [20] can't broadcast.
    ```

    Now let's consider batches of inputs in conjunction with batches of kernel
    parameters. We require that the input batch shapes be broadcastable with
    the kernel parameter batch shapes, otherwise we get an error:

    ```python
    x = np.ones([10, 5, 6, 3], np.float32)
    y = np.ones([10, 7, 8, 3], np.float32)

    batch_kernel = tfp.math.psd_kernels.SomeKernel(params=[1., .5])
    batch_kernel.batch_shape
    # ==> [2]
    batch_kernel.tensor(x, y, x1_example_ndims=2, x2_example_ndims=2).shape
    # ==> Error! [2] and [10] can't broadcast.
    ```

    The fix is to make the kernel parameter shape broadcastable with `[10]` (or
    reshape the inputs to be broadcastable!):

    ```python
    x = np.ones([10, 5, 6, 3], np.float32)
    y = np.ones([10, 7, 8, 3], np.float32)

    batch_kernel = tfp.math.psd_kernels.SomeKernel(
        params=[[1.], [.5]])
    batch_kernel.batch_shape
    # ==> [2, 1]
    batch_kernel.tensor(x, y, x1_example_ndims=2, x2_example_ndims=2).shape
    # ==> [2, 10, 5, 6, 7, 8]

    # Or, make the inputs broadcastable:
    x = np.ones([10, 1, 5, 6, 3], np.float32)
    y = np.ones([10, 1, 7, 8, 3], np.float32)

    batch_kernel = tfp.math.psd_kernels.SomeKernel(
        params=[1., .5])
    batch_kernel.batch_shape
    # ==> [2]
    batch_kernel.tensor(x, y, x1_example_ndims=2, x2_example_ndims=2).shape
    # ==> [10, 2, 5, 6, 7, 8]
    ```

    """
    with self._name_and_control_scope(self._name):
      x1 = tf.convert_to_tensor(x1, name='x1', dtype_hint=self.dtype)
      x2 = tf.convert_to_tensor(x2, name='x2', dtype_hint=self.dtype)

      x1 = util.pad_shape_with_ones(
          x1,
          ndims=x2_example_ndims,
          start=-(self.feature_ndims + 1))

      x2 = util.pad_shape_with_ones(
          x2,
          ndims=x1_example_ndims,
          start=-(self.feature_ndims + 1 + x2_example_ndims))

      return self.apply(
          x1, x2, example_ndims=(x1_example_ndims + x2_example_ndims))

  def _batch_shape(self):
    raise NotImplementedError('Subclasses must provide batch_shape property.')

  def _batch_shape_tensor(self):
    raise NotImplementedError(
        'Subclasses must provide batch_shape_tensor implementation')

  def __add__(self, k):
    if not isinstance(k, PositiveSemidefiniteKernel):
      raise ValueError(
          "Can't add non-kernel (of type '%s') to kernel" % type(k))
    return _SumKernel([self, k])

  def __iadd__(self, k):
    return self.__add__(k)

  def __mul__(self, k):
    if not isinstance(k, PositiveSemidefiniteKernel):
      raise ValueError(
          "Can't multiply by non-kernel (of type '%s') to kernel" % type(k))
    return _ProductKernel([self, k])

  def __imul__(self, k):
    return self.__mul__(k)

  def __str__(self):
    return ('tfp.math.psd_kernels.{type_name}('
            '"{self_name}"'
            '{maybe_batch_shape}'
            ', feature_ndims={feature_ndims}'
            ', dtype={dtype})'.format(
                type_name=type(self).__name__,
                self_name=self.name,
                maybe_batch_shape=(', batch_shape={}'.format(self.batch_shape)
                                   if self.batch_shape.ndims is not None else
                                   ''),
                feature_ndims=self.feature_ndims,
                dtype=None if self.dtype is None else self.dtype.name))

  def __repr__(self):
    return ('<tfp.math.psd_kernels.{type_name} '
            '\'{self_name}\''
            ' batch_shape={batch_shape}'
            ' feature_ndims={feature_ndims}'
            ' dtype={dtype}>'.format(
                type_name=type(self).__name__,
                self_name=self.name,
                batch_shape=self.batch_shape,
                feature_ndims=self.feature_ndims,
                dtype=None if self.dtype is None else self.dtype.name))

  def _parameter_control_dependencies(self, is_init):
    """Returns a list of ops to be executed in members with graph deps.

    Typically subclasses override this function to return parameter specific
    assertions (eg, positivity of `amplitude`, etc.).

    Args:
      is_init: Python `bool` indicating that the call site is `__init__`.

    Returns:
      dependencies: `list`-like of ops to be executed in member functions with
        graph dependencies.
    """
    return ()


def _flatten_summand_list(kernels):
  """Flatten a list of kernels which may contain _SumKernel instances.

  Args:
    kernels: Python list of `PositiveSemidefiniteKernel` instances

  Returns:
    Python list containing the elements of kernels, with any _SumKernel
    instances replaced by their `kernels` property contents.
  """
  flattened = []
  for k in kernels:
    if isinstance(k, _SumKernel):
      flattened += k.kernels
    else:
      flattened.append(k)
  return flattened


def _flatten_multiplicand_list(kernels):
  """Flatten a list of kernels which may contain _ProductKernel instances.

  Args:
    kernels: Python list of `PositiveSemidefiniteKernel` instances

  Returns:
    Python list containing the elements of kernels, with any _ProductKernel
    instances replaced by their `kernels` property contents.
  """
  flattened = []
  for k in kernels:
    if isinstance(k, _ProductKernel):
      flattened += k.kernels
    else:
      flattened.append(k)
  return flattened


class _SumKernel(PositiveSemidefiniteKernel):
  """Kernel class representing summation over a list of kernels.

  Mathematically this class represents the pointwise sum of several kernels.
  Given two kernels, `k1` and `k2`, and `kp = _SumKernel([k1, k2])`, we have

    ```none
    kp.apply(x, y) = k1(x, y) + k2(x, y)
    ```

  for any `x`, `y` in the feature space (this presumes that the constituent
  kernels all act on the same feature space).

  That the sum is positive semi-definite follows simply from the definition of
  positive semi-definiteness of functions. If we have

    ```none
    sum_i sum_j (c[i]*) c[j] k1(x[i], x[j]) >= 0
    ```
  and

    ```none
    sum_i sum_j (c[i]*) c[j] k2(x[i], x[j]) >= 0
    ```

  for any finite collections `{x[1], ..., x[N]}` in S and `{c[1], ..., c[N]}` in
  the reals (or the complex plane), then we clearly also have the same for the
  sum of `k1` and `k2`.
  """

  def __init__(self, kernels, name=None):
    """Create a kernel which is the sum of `kernels`.

    The input list is 'flattened' in the sense that any entries which are also
    of type `_SumKernel` will have their list of kernels appended to this
    instance's list of kernels. This will reduce the stack depth when actually
    evaluating the sum over kernel applications.

    Args:
      kernels: Python `list` of `PositiveSemidefiniteKernel` instances.
      name: Python `str` name prefixed to Ops created by this class.
    Raises:
      ValueError: `kernels` is an empty list, or `kernels` don't all have the
      same `feature_ndims`.
    """
    if not kernels:
      raise ValueError("Can't create _SumKernel over empty list.")
    if len(set([k.feature_ndims for k in kernels])) > 1:
      raise ValueError(
          "Can't sum kernels with different feature_ndims. Got:\n%s" %
          str([k.feature_ndims for k in kernels]))
    self._kernels = _flatten_summand_list(kernels)
    if name is None:
      name = 'SumKernel'
    # We have ensured the list is non-empty and all feature_ndims are the same.
    super(_SumKernel, self).__init__(
        feature_ndims=kernels[0].feature_ndims,
        dtype=util.maybe_get_common_dtype(
            [None if k.dtype is None else k for k in kernels]),
        name=name,
        validate_args=any([k.validate_args for k in kernels]))

  @property
  def kernels(self):
    """The list of kernels this _SumKernel sums over."""
    return self._kernels

  def _apply(self, x1, x2, example_ndims=0):
    return sum([k.apply(x1, x2, example_ndims) for k in self.kernels])

  def _batch_shape(self):
    return functools.reduce(tf.broadcast_static_shape,
                            [k.batch_shape for k in self.kernels])

  def _batch_shape_tensor(self):
    return functools.reduce(tf.broadcast_dynamic_shape,
                            [k.batch_shape_tensor() for k in self.kernels])


class _ProductKernel(PositiveSemidefiniteKernel):
  """Kernel class representing the product over a list of kernels.

  Mathematically this class represents the pointwise product of several kernels.
  Given two kernels, `k1` and `k2`, and `kp = _ProductKernel([k1, k2])`, we have

    ```none
    kp.apply(x, y) = k1(x, y) * k2(x, y)
    ```

  for any x, y in the feature space (this presumes that the constituent kernels
  all act on the same feature space).

  The fact that this product is still positive semi-definite can be shown in a
  variety of ways, many deep and all fascinating, but follows readily from the
  [Schur product theorem](https://en.wikipedia.org/wiki/Schur_product_theorem),
  which states that the Hadamard (element-wise) product of two PSD matrices is
  also PSD.
  """

  def __init__(self, kernels, name=None):
    """Create a kernel which is the product of `kernels`.

    The input list is 'flattened' in the sense that any entries which are also
    of type `_ProductKernel` will have their list of kernels appended to this
    instance's list of kernels. This will reduce the stack depth when actually
    evaluating the product over kernel applications.

    Args:
      kernels: Python `list` of `PositiveSemidefiniteKernel` instances.
      name: Python `str` name prefixed to Ops created by this class.
    Raises:
      ValueError: `kernels` is an empty list, or `kernels` don't all have the
      same `feature_ndims`.
    """
    if not kernels:
      raise ValueError("Can't create _ProductKernel over empty list.")
    if len(set([k.feature_ndims for k in kernels])) > 1:
      raise ValueError(
          "Can't multiply kernels with different feature_ndims. Got:\n%s" %
          str([k.feature_ndims for k in kernels]))
    self._kernels = _flatten_multiplicand_list(kernels)
    if name is None:
      name = 'ProductKernel'
    # We have ensured the list is non-empty and all feature_ndims are the same.
    super(_ProductKernel, self).__init__(
        feature_ndims=kernels[0].feature_ndims,
        dtype=util.maybe_get_common_dtype(
            [None if k.dtype is None else k for k in kernels]),
        name=name,
        validate_args=any([k.validate_args for k in kernels]))

  @property
  def kernels(self):
    """The list of kernels this _ProductKernel multiplies over."""
    return self._kernels

  def _apply(self, x1, x2, example_ndims=0):
    return functools.reduce(
        operator.mul,
        [k.apply(x1, x2, example_ndims) for k in self.kernels])

  def _batch_shape(self):
    return functools.reduce(tf.broadcast_static_shape,
                            [k.batch_shape for k in self.kernels])

  def _batch_shape_tensor(self):
    return functools.reduce(tf.broadcast_dynamic_shape,
                            [k.batch_shape_tensor() for k in self.kernels])