# Linear¶

class probnum.randprocs.kernels.Linear(input_shape, constant=0.0)

Linear kernel.

Linear covariance function defined by

$k(x_0, x_1) = x_0^\top x_1 + c.$
Parameters

Polynomial

Polynomial covariance function.

Examples

>>> import numpy as np
>>> from probnum.randprocs.kernels import Linear
>>> K = Linear(input_shape=2)
>>> xs = np.array([[1, 2], [2, 3]])
>>> K.matrix(xs)
array([[ 5.,  8.],
[ 8., 13.]])


Attributes Summary

 input_ndim Syntactic sugar for len(input_shape). input_shape Shape of single, i.e. non-batched, arguments of the covariance function. output_ndim Syntactic sugar for len(output_shape). output_shape Shape of single, i.e. non-batched, return values of the covariance function.

Methods Summary

 __call__(x0, x1) Evaluate the (cross-)covariance function(s). matrix(x0[, x1]) A convenience function for computing a kernel matrix for two sets of inputs.

Attributes Documentation

input_ndim

Syntactic sugar for len(input_shape).

Return type

int

input_shape

Shape of single, i.e. non-batched, arguments of the covariance function.

Return type
output_ndim

Syntactic sugar for len(output_shape).

Return type

int

output_shape

Shape of single, i.e. non-batched, return values of the covariance function.

If output_shape is (), the Kernel instance represents a single (cross-)covariance function. Otherwise, i.e. if output_shape is non-empty, the Kernel instance represents a tensor of (cross-)covariance functions whose shape is given by output_shape.

Return type

Methods Documentation

__call__(x0, x1)

Evaluate the (cross-)covariance function(s).

The evaluation of the (cross-covariance) function(s) is vectorized over the batch shapes of the arguments, applying standard NumPy broadcasting.

Parameters
Returns

shape= bcast_batch_shape + output_shape – The (cross-)covariance function(s) evaluated at (x0, x1). Since the function is vectorized over the batch shapes of the inputs, the output array contains the following entries:

k_x0_x1[batch_idx + output_idx] = k[output_idx](
x0[batch_idx, ...],
x1[batch_idx, ...],
)


where we assume that x0 and x1 have been broadcast to a common shape bcast_batch_shape + input_shape, and where output_idx and batch_idx are indices compatible with output_shape and bcast_batch_shape, respectively. By k[output_idx] we refer to the covariance function at index output_idx in the tensor of covariance functions represented by the Kernel instance.

Return type

k_x0_x1

Raises

matrix

Convenience function to compute a kernel matrix, i.e. a matrix of pairwise evaluations of the kernel on two sets of points.

Examples

See documentation of class Kernel.

matrix(x0, x1=None)

A convenience function for computing a kernel matrix for two sets of inputs.

This is syntactic sugar for k(x0[:, None], x1[None, :]). Hence, it computes the matrix (stack) of pairwise covariances between two sets of input points. If k represents a single covariance function, then the resulting matrix will be symmetric positive-(semi)definite for x0 == x1.

Parameters
Returns

shape= batch_shape + output_shape – The matrix / stack of matrices containing the pairwise evaluations of the (cross-)covariance function(s) on x0 and x1. Depending on the shape of the inputs, batch_shape is either (M, N), (M,), (N,), or ().

Return type

kernmat

Raises

ValueError – If the shapes of the inputs don’t match the specification.

__call__
See documentation of class Kernel.