code: # 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.compat.v2 as tf
  
  from tensorflow_probability.python.positive_semidefinite_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.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]
  
      scalar_kernel.matrix(x, y).shape
      # ==> [5, 5]
      ```
  
      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]
  
      # `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):
      """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, 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
  
    @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(
              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) as scope:
          yield 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.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 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.positive_semidefinite_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.positive_semidefinite_kernels.SomeKernel(param=[.2, .5])
      batch_kernel.batch_shape
      # ==> [2]
      batch_kernel.apply(x, y, example_ndims=1).shape
      # ==> [2, 5]
  
      """
      with self._name_scope(self._name, values=[x1, x2]):
        x1 = tf.convert_to_tensor(x1, name='x1')
        x2 = tf.convert_to_tensor(x2, name='x2')
  
        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.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(x1, name='x1')
        x2 = tf.convert_to_tensor(x2, name='x2')
  
        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.positive_semidefinite_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.positive_semidefinite_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.positive_semidefinite_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.positive_semidefinite_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.positive_semidefinite_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_scope(self._name, values=[x1, x2]):
        x1 = tf.convert_to_tensor(x1, name='x1')
        x2 = tf.convert_to_tensor(x2, name='x2')
  
        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.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=None if self.dtype is None else 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=None if self.dtype is None else 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,
          dtype=util.maybe_get_common_dtype(
              [None if k.dtype is None else k for k in kernels]),
          name=name)
  
    @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)
  
    @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])
   # 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.
  # You may obtain a copy of the License at
  #
  #     http://www.apache.org/licenses/LICENSE-2.0
  #
  # Unless required by applicable law or agreed to in writing, software
  # distributed under the License is distributed on an "AS IS" BASIS,
  # WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
  # See the License for the specific language governing permissions and
  # limitations under the License.
  # ============================================================================
  """Numpy implementations of TensorFlow functions."""
  
  from __future__ import absolute_import
  from __future__ import division
  from __future__ import print_function
  
  # Dependency imports
  import numpy as np
  
  import tensorflow as tf
  
  from tensorflow_probability.python.internal.backend.numpy import numpy_array
  
  from tensorflow_probability.python.internal.backend.numpy.internal import utils
  from tensorflow_probability.python.internal.backend.numpy.numpy_math import l2_normalize
  from tensorflow_probability.python.internal.backend.numpy.numpy_math import log_softmax
  from tensorflow_probability.python.internal.backend.numpy.numpy_math import reduce_logsumexp
  from tensorflow_probability.python.internal.backend.numpy.numpy_math import softmax
  from tensorflow_probability.python.internal.backend.numpy.numpy_math import softplus
  from tensorflow_probability.python.internal.backend.numpy.numpy_math import top_k
  
  
  __all__ = [
      'l2_normalize',
      'log_softmax',
      'relu',
      'softmax',
      'softplus',
      'sigmoid_cross_entropy_with_logits',
      'sparse_softmax_cross_entropy_with_logits',
      'top_k',
  ]
  
  
  def _sigmoid_cross_entropy_with_logits(  # pylint: disable=invalid-name,unused-argument
      _sentinel=None,
      labels=None,
      logits=None,
      name=None):
    return (np.maximum(logits, 0)
            - logits * labels + np.log1p(np.exp(-np.abs(logits))))
  
  
  def _sparse_softmax_cross_entropy_with_logits(  # pylint: disable=invalid-name,unused-argument
      _sentinel=None,
      labels=None,
      logits=None,
      name=None):
    """Softmax cross entropy with logits."""
    labels_shape = labels.shape
    num_classes = logits.shape[-1]
    logits = np.reshape(logits, [-1, num_classes])
    labels = np.reshape(labels, [-1])
  
    labels = numpy_array.one_hot(labels, num_classes)
  
    cost = -np.sum(
        labels * (logits - reduce_logsumexp(logits, axis=-1, keepdims=True)),
        axis=-1)
    cost = np.reshape(cost, labels_shape)
    return cost
  
  
  # --- Begin Public Functions --------------------------------------------------
  
  l2_normalize = utils.copy_docstring(
      tf.nn.l2_normalize,
      l2_normalize)
  
  
  relu = utils.copy_docstring(
      tf.nn.relu,
      lambda features, name=None: np.max(features, 0))
  
  
  softplus = utils.copy_docstring(
      tf.nn.softplus,
      lambda features, name=None: np.log(1 + np.exp(features)))
  
  
  sigmoid_cross_entropy_with_logits = utils.copy_docstring(
      tf.nn.sigmoid_cross_entropy_with_logits,
      _sigmoid_cross_entropy_with_logits)
  
  
  sparse_softmax_cross_entropy_with_logits = utils.copy_docstring(
      tf.nn.sparse_softmax_cross_entropy_with_logits,
      _sparse_softmax_cross_entropy_with_logits)
  
apis: tensorflow.compat.v2.name_scope tensorflow.compat.v2.squeeze tensorflow.compat.v2.expand_dims tensorflow.compat.v2.convert_to_tensor
apis: numpy.reshape numpy.abs numpy.exp numpy.max numpy.maximum