tfp.substrates.jax.math.psd_kernels.Polynomial

Polynomial Kernel.

Inherits From: AutoCompositeTensorPsdKernel, PositiveSemidefiniteKernel

tfp.substrates.jax.math.psd_kernels.Polynomial(
    bias_amplitude=None,
    slope_amplitude=None,
    shift=None,
    exponent=None,
    feature_ndims=1,
    validate_args=False,
    parameters=None,
    name='Polynomial'
)

Is based on the dot product covariance function and can be obtained from polynomial regression. This kernel, when parameterizing a Gaussian Process, results in random polynomial functions. A linear kernel can be created from this by setting the exponent to 1 or None.

k(x, y) = bias_amplitude**2 + slope_amplitude**2 *
          ((x - shift) dot (y - shift))**exponent

References

[1]: Carl Edward Rasmussen and Christopher K. I. Williams. Gaussian Processes for Machine Learning. Section 4.4.2. 2006. http://www.gaussianprocess.org/gpml/chapters/RW4.pdf [2]: David Duvenaud. The Kernel Cookbook. https://www.cs.toronto.edu/~duvenaud/cookbook/

bias_amplitude Non-negative floating point Tensor that controls the stddev from the origin. If bias = 0, there is no stddev and the fitted function goes through the origin. Must be broadcastable with slope_amplitude, shift, exponent, and inputs to apply and matrix methods. A value of None is treated like 0. Default Value: None
slope_amplitude Non-negative floating point Tensor that controls the stddev of the regression line slope that is the basis for the polynomial. Must be broadcastable with bias_amplitude, shift, exponent, and inputs to apply and matrix methods. A value of None is treated like 1. Default Value: None
shift Floating point Tensor that contols the intercept with the x-axis of the linear function to be exponentiated to get this polynomial. Must be broadcastable with bias_amplitude, slope_amplitude, exponent and inputs to apply and matrix methods. A value of None is treated like 0, which results in having the intercept at the origin. Default Value: None
exponent Positive floating point Tensor that controls the exponent (also known as the degree) of the polynomial function, and must be an integer. Must be broadcastable with bias_amplitude, slope_amplitude, shift, and inputs to apply and matrix methods. A value of None is treated like 1, which results in a linear kernel. Default Value: None
feature_ndims Python int number of rightmost dims to include in kernel computation. Default Value: 1
validate_args If True, parameters are checked for validity despite possibly degrading runtime performance. Default Value: False
parameters For subclasses, a dict of constructor arguments.
name Python str name prefixed to Ops created by this class. Default Value: 'Polynomial'

batch_shape Shape of a single sample from a single event index as a TensorShape.

May be partially defined or unknown.

The batch dimensions are indexes into independent, non-identical parameterizations of this PositiveSemidefiniteKernel.

bias_amplitude Stddev on bias parameter.
dtype (Nested) dype over which the kernel operates.
exponent Exponent of the polynomial term.
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. Inputs may also be nested structures, in which case feature_ndims is a parallel nested structure containing the feature rank of each component.

name Name prepended to all ops created by this class.
parameters Dictionary of parameters used to instantiate this PSDKernel.
shift Shift of linear function that is exponentiated.
slope_amplitude Amplitude on slope parameter.
trainable_variables

validate_args Python bool indicating possibly expensive checks are enabled.
variables

Methods

apply

View source

apply(
    x1, x2, example_ndims=0, name='apply'
)

Apply the kernel function pairs of inputs.

Args
x1 (Nested) Tensor input to the kernel, of shape B1 + E1 + F, where B1 and E1 may be empty (ie, no batch/example dims, resp.). If nested, B1 and E1 must broadcast across elements of the structure. F (the feature shape) must have rank equal to the kernel's feature_ndims property, or to the corresponding element of the feature_ndims nested structure. 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 (Nested) Tensor input to the kernel, of shape B2 + E2 + F, where B2 and E2 may be empty (ie, no batch/example dims, resp.). If nested, B1 and E1 must broadcast across elements of the structure. F (the feature shape) must have rank equal to the kernel's feature_ndims property, or to the corresponding element of the feature_ndims nested structure. 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).
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.
name name to give to the op

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:

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

import tensorflow_probability as tfp; tfp = tfp.substrates.jax

# 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

[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.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:

Fix #1

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

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

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]

batch_shape_tensor

View source

batch_shape_tensor(
    name='batch_shape_tensor'
)

Shape of a single sample from a single event index as a 1-D Tensor.

The batch dimensions are indexes into independent, non-identical parameterizations of this PositiveSemidefiniteKernel.

Args
name name to give to the op

Returns
batch_shape Tensor.

copy

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copy(
    **override_parameters_kwargs
)

Creates a copy of the kernel.

Args
**override_parameters_kwargs String/value dictionary of initialization arguments to override with new values.

Returns
copied_kernel A new instance of type(self) initialized from the union of self.parameters and override_parameters_kwargs, i.e., dict(self.parameters, **override_parameters_kwargs).

matrix

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matrix(
    x1, x2, name='matrix'
)

Construct (batched) matrices from (batches of) collections of inputs.

Args
x1 (Nested) 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 (or to the corresponding element of feature_ndims, if nested). Batch shape must broadcast with the batch shape of x2 and with the kernel's batch shape.
x2 (Nested) 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 (or to the corresponding element of feature_ndims, if nested). Batch shape must broadcast with the batch shape of x1 and with the kernel's batch shape.
name name to give to the op

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

[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).

Examples

First, consider a kernel with a single scalar parameter.

import tensorflow_probability as tfp; tfp = tfp.substrates.jax

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:

  | 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.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.

# 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.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!):

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.

parameter_properties

View source

@classmethod
parameter_properties(
    dtype=tf.float32
)

Returns a dict mapping constructor arg names to property annotations.

This dict should include an entry for each of the kernel's Tensor-valued constructor arguments.

Args
dtype Optional float dtype to assume for continuous-valued parameters. Some constraining bijectors require advance knowledge of the dtype because certain constants (e.g., tfb.Softplus.low) must be instantiated with the same dtype as the values to be transformed.

Returns
parameter_properties A str ->tfp.python.internal.parameter_properties.ParameterPropertiesdict mapping constructor argument names toParameterProperties` instances.

tensor

View source

tensor(
    x1, x2, x1_example_ndims, x2_example_ndims, name='tensor'
)

Construct (batched) tensors from (batches of) collections of inputs.

Args
x1 (Nested) 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 (or to the corresponding element of feature_ndims, if nested). Batch shape must broadcast with the batch shape of x2 and with the kernel's batch shape.
x2 (Nested) 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 (or to the corresponding element of feature_ndims, if nested). 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.
name name to give to the op

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.

import tensorflow_probability as tfp; tfp = tfp.substrates.jax

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:

  | 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.

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.

# 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:

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:

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!):

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]

__add__

View source

__add__(
    k
)

__getitem__

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__getitem__(
    slices
)

Slices the batch axes of this kernel, returning a new instance.

k = tfpk.ExponentiatedQuadratic(
  amplitude=tf.ones([3, 5, 7, 9]),
  length_scale=tf.ones([3, 5, 7, 9]))
k.batch_shape  # => [3, 5, 7, 9]
k2 = k[:, tf.newaxis, ..., -2:, 1::2]
k2.batch_shape  # => [3, 1, 5, 2, 4]

Args
slices slices from the [] operator

Returns
dist A new PositiveSemidefiniteKernel instance with sliced parameters.

__iter__

View source

__iter__()

__mul__

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__mul__(
    k
)

__radd__

View source

__radd__(
    lhs
)