# Copyright 2018 The TensorFlow Probability Authors.
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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 as tf

__all__ = [
    'PositiveSemidefiniteKernel',
]


@six.add_metaclass(abc.ABCMeta)
class PositiveSemidefiniteKernel(object):
  """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 implementing 2 primary public methods:
  `matrix` and `apply`.

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

  `apply` computes the value of the kernel function at a pair of (batches of)
  input locations. It is the more low-level operation and must be implemented in
  each concrete base class of PositiveSemidefiniteKernel.

  #### Kernel Parameter Shape Semantics

  PositiveSemidefiniteKernel implementations support batching of kernel
  parameters. 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

  `apply` and `matrix` 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, e, f1, ..., fF]
  '----------'  |  '---------'
       |        |       '-- Feature dimensions
       |        '-- Example dimension (`matrix`-only)
       '-- 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.
  - Example dimensions are relevant only for `matrix` calls. Given inputs `x`
    and `y` with feature dimensions [f1, ..., fF] and example dimensions `e1`
    and `e2`, a to `matrix` will yield an `e1 x e2` matrix. If batch dimensions
    are present, it will return a batch of `e1 x e2` matrices.
  - Batch dimensions are supported for inputs to `apply` or `matrix`; the only
    requirement is that batch dimensions of inputs `x` and `y` be broadcastable
    with each other and with the kernel's parameter batch shapes (see above).
  """

  def __init__(self, feature_ndims, dtype=None, name=None):
    """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.

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

    ```none
    [b1, ..., bB, e, f1, ..., fF]
    '----------'  |  '---------'
         |        |       '-- Feature dimensions
         |        '-- Example dimension (`matrix`-only)
         '-- 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

  @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 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(
            value=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_scope(self, name=None, values=None):
    """Helper function to standardize op scope."""
    with tf.name_scope(self.name):
      values = [] if values is None else values
      with tf.name_scope(name, values=values) as scope:
        yield scope

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

    Args:
      x1: `Tensor` input to the first positional parameter of the kernel, of
        shape `[b1, ..., bB, f1, ..., fF]`, where `B` may be zero (ie, no
        batching) and `F` (number of feature dimensions) must equal the kernel's
        `feature_ndims` property. Batch shape must broadcast with the batch
        shape of `x2` and with the kernel's parameters.
      x2: `Tensor` input to the second positional parameter of the kernel,
        shape `[c1, ..., cC, f1, ..., fF]`, where `C` may be zero (ie, no
        batching) and `F` (number of feature dimensions) must equal the kernel's
        `feature_ndims` property. Batch shape must broadcast with the batch
        shape of `x1` and with the kernel's parameters.

    Returns:
      `Tensor` containing the (batch of) results of applying the kernel function
      to inputs `x1` and `x2`. If the kernel parameters' batch shape is
      `[k1, ..., kK]` then the shape of the `Tensor` resulting from this method
      call is `broadcast([b1, ..., bB], [c1, ..., cC], [k1, ..., kK])`.

    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. Given the same kernel and, say, batched inputs of shape
    `[b1, ..., bB, f1, ..., fF]`, it will yield a batch of scalars of shape
    `[b1, ..., bB]`.

    #### Examples

    ```python
    import tensorflow_probability as tfp

    # Suppose `SomeKernel` acts on vectors (rank-1 tensors)
    scalar_kernel = tfp.positive_semidefinite_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.positive_semidefinite_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 by giving the parameter a shape of
    `[2, 1]` which will correctly broadcast with `[5]` to yield `[2, 5]`:

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

    """
    with self._name_scope(self._name, values=[x1, x2]):
      x1 = tf.convert_to_tensor(value=x1, name='x1')
      x2 = tf.convert_to_tensor(value=x2, name='x2')
      return self._apply(x1, x2)

  def _apply(self, x1, x2, param_expansion_ndims=0):
    """Apply the kernel function to a pair of (batches of) inputs.

    Args:
      x1: `Tensor` input to the first positional parameter of the kernel, of
        shape `[b1, ..., bB, f1, ..., fF]`, where `B` may be zero (no batching)
        and `F` (number of feature dimensions) must equal the kernel's
        `feature_ndims` property. Batch shape must broadcast with the batch
        shape of `x2` and with the kernel's parameters *after* parameter
        expansion (see `param_expansion_ndims` argument).
      x2: `Tensor` input to the second positional parameter of the kernel,
        shape `[c1, ..., cC, f1, ..., fF]`, where `C` may be zero (no batching)
        and `F` (number of feature dimensions) must equal the kernel's
        `feature_ndims` property. Batch shape must broadcast with the batch
        shape of `x1` and with the kernel's parameters *after* parameter
        expansion (see `param_expansion_ndims` argument).
      param_expansion_ndims: Python `integer` enabling reshaping of kernel
        parameters by concatenating a list of 1's to the param shapes. This
        allows the caller to control how the parameters broadcast across the
        inputs.

    Returns:
      `Tensor` containing the (batch of) results of applying the kernel function
      to inputs `x1` and `x2`. If the kernel parameters' batch shape *after*
      parameter expansion (ie, concatenating `param_expansion_ndims` 1's onto
      the parameters' shapes) is `[k1, ..., kK, 1, ..., 1]` then the shape of
      the `Tensor` resulting from this method call is
      `broadcast([b1, ..., bB], [c1, ..., cC], [k1, ..., kK, 1, ..., 1])`.
    """
    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, ..., bB, e1, f1, ..., fF]`, where `B` may be zero (ie, no
        batching), e1 is an integer greater than zero, and `F` (number of
        feature dimensions) must equal the kernel's `feature_ndims` property.
        Batch shape must broadcast with the batch shape of `x2` and with the
        kernel's parameters *after* parameter expansion (see
        `param_expansion_ndims` argument).
      x2: `Tensor` input to the second positional parameter of the kernel,
        shape `[c1, ..., cC, e2, f1, ..., fF]`, where `C` may be zero (ie, no
        batching), e2 is an integer greater than zero,  and `F` (number of
        feature dimensions) must equal the kernel's `feature_ndims` property.
        Batch shape must broadcast with the batch shape of `x1` and with the
        kernel's parameters *after* parameter expansion (see
        `param_expansion_ndims` argument).

    Returns:
      `Tensor containing (batch of) matrices of kernel applications to pairs
      from inputs `x1` and `x2`. If the kernel parameters' batch shape is
      `[k1, ..., kK]`, then the shape of the resulting `Tensor` is
      `broadcast([b1, ..., bB], [c1, ..., cC], [k1, ..., kK]) + [e1, e2]`.

    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).

    N.B., this method can only be used to compute the pairwise application of
    the kernel function on rank-1 collections. E.g., it *does* support inputs of
    shape `[e1, f]` and `[e2, f]`, yielding a matrix of shape `[e1, e2]`. It
    *does not* support inputs of shape `[e1, e2, f]` and `[e3, e4, f]`, yielding
    a `Tensor` of shape `[e1, e2, e3, e4]`. To do this, one should instead
    reshape the inputs and pass them to `apply`, e.g.:

    ```python
    k = psd_kernels.SomeKernel()
    t1 = tf.placeholder([4, 4, 3], tf.float32)
    t2 = tf.placeholder([5, 5, 3], tf.float32)
    k.apply(
        tf.reshape(t1, [4, 4, 1, 1, 3]),
        tf.reshape(t2, [1, 1, 5, 5, 3])).shape
    # ==> [4, 4, 5, 5, 3]
    ```

    `matrix` is a special case of the above, where there is only one example
    dimension; indeed, its implementation looks almost exactly like the above
    (reshaped inputs passed to the private version of `_apply`).

    #### Examples

    First, consider a kernel with a single scalar parameter.

    ```python
    import tensorflow_probability as tfp

    scalar_kernel = tfp.positive_semidefinite_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.positive_semidefinite_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.positive_semidefinite_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.positive_semidefinite_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.positive_semidefinite_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_scope(self._name, values=[x1, x2]):
      x1 = tf.convert_to_tensor(value=x1, name='x1')
      x2 = tf.convert_to_tensor(value=x2, name='x2')
      x1 = tf.expand_dims(x1, -(self.feature_ndims + 1))
      x2 = tf.expand_dims(x2, -(self.feature_ndims + 2))
      return self._apply(x1, x2, param_expansion_ndims=2)

  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.positive_semidefinite_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=self.dtype.name))

  def __repr__(self):
    return ('<tfp.positive_semidefinite_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=self.dtype.name))


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,
                                     name=name)

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

  def _apply(self, x1, x2, param_expansion_ndims=0):
    return sum([k._apply(x1, x2, param_expansion_ndims) for k in self.kernels])  # pylint: disable=protected-access

  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,
                                         name=name)

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

  def _apply(self, x1, x2, param_expansion_ndims=0):
    return functools.reduce(
        operator.mul,
        [k._apply(x1, x2, param_expansion_ndims) for k in self.kernels])  # pylint: disable=protected-access

  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])
