tfp.positive_semidefinite_kernels.PositiveSemidefiniteKernel

Class PositiveSemidefiniteKernel

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

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:

[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).

__init__

__init__(
    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:
[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.

Properties

batch_shape

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:

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

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.

dtype

DType over which the kernel operates.

feature_ndims

The number of feature dimensions.

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

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.

name

Name prepended to all ops created by this class.

Methods

__add__

__add__(k)

__iadd__

__iadd__(k)

__imul__

__imul__(k)

__mul__

__mul__(k)

apply

apply(
    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:

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

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

[k(x[0], y[0]), k(x[1], y[1]), ..., k(x[4], y[4])]

Now we can consider a kernel with batched parameters:

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]:

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

batch_shape_tensor

batch_shape_tensor()

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.

matrix

matrix(
    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 inputsx1andx2. If the kernel parameters' batch shape is[k1, ..., kK], then the shape of the resultingTensorisbroadcast([b1, ..., bB], [c1, ..., cC], [k1, ..., kK]) + [e1, e2]`.

Given inputs x1 and x2 of shapes

[b1, ..., bB, e1, f1, ..., fF]

and

[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.:

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.

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:

| 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

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.

# 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:

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:

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!):

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.