# Kernel¶

class probnum.kernels.Kernel(input_dim, output_dim=1)

Bases: typing.Generic, abc.ABC

Kernel / covariance function.

Abstract base class for kernels / covariance functions. Kernels are a generalization of a positive-definite function or matrix. They typically define the covariance function of a random process and thus describe its spatial or temporal variation. If evaluated at two sets of points a kernel gives the covariance of the random process at these locations.

Parameters
• input_dim (Integral) – Input dimension of the kernel.

• output_dim (Integral) – Output dimension of the kernel.

Examples

Kernels are implemented by subclassing this abstract base class.

>>> from probnum.kernels import Kernel
...
>>> class CustomLinearKernel(Kernel):
...
...     def __init__(self, constant=0.0):
...         self.constant = constant
...         super().__init__(input_dim=1, output_dim=1)
...
...     def __call__(self, x0, x1=None):
...         # Check and reshape inputs
...         x0, x1, kernshape = self._check_and_reshape_inputs(x0, x1)
...
...         # Compute kernel matrix
...         if x1 is None:
...             x1 = x0
...         kernmat = x0 @ x1.T + self.constant
...
...         return Kernel._reshape_kernelmatrix(kernmat, newshape=kernshape)

We can now evaluate the kernel like so.

>>> import numpy as np
>>> k = CustomLinearKernel(constant=1.0)
>>> k(np.linspace(0, 1, 4)[:, None])
array([[1.        , 1.        , 1.        , 1.        ],
[1.        , 1.11111111, 1.22222222, 1.33333333],
[1.        , 1.22222222, 1.44444444, 1.66666667],
[1.        , 1.33333333, 1.66666667, 2.        ]])

Attributes Summary

 input_dim Dimension of arguments of the covariance function. output_dim Dimension of the evaluated covariance function.

Methods Summary

 __call__(x0[, x1]) Evaluate the kernel.

Attributes Documentation

input_dim

Dimension of arguments of the covariance function.

The dimension of inputs to the covariance function $$k : \mathbb{R}^{ d_{in}} \times \mathbb{R}^{d_{in}} \rightarrow \mathbb{R}^{d_{out} \times d_{out}}$$.

Return type

int

output_dim

Dimension of the evaluated covariance function.

The resulting evaluated kernel $$k(x_0, x_1) \in \mathbb{R}^{d_{out} \times d_{out}}$$ has shape=(output_dim, output_dim).

Return type

int

Methods Documentation

abstract __call__(x0, x1=None)[source]

Evaluate the kernel.

Computes the covariance function at x0 and x1. If the inputs have more than one dimension the covariance function is evaluated pairwise for all observations determined by the first dimension of x0 and x1. If only x0 is given the kernel matrix $$K=k(X_0, X_0)$$ is computed.

Parameters
• x0 (~InputType) – shape=(input_dim,) or (n0, input_dim) – First input.

• x1 (Optional[~InputType]) – shape=(input_dim,) or (n1, input_dim) – Second input.

Returns

shape=(), (output_dim, output_dim) or (n0, n1) or (n0, n1, output_dim, output_dim) – Kernel evaluated at x0 and x1 or kernel matrix containing pairwise evaluations for all observations in x0 (and x1).

Return type

cov