jaxkern-nightly 0.0.5.dev20230423

Kernels in Jax.

Kernels in Jax.

Introduction

JaxKern is Python library for working with kernel functions in JAX. We currently support the following kernels:

• Stationary
  • Radial basis function (Squared exponential)
  • Matérn
  • Powered exponential
  • Rational quadratic
  • White noise
  • Periodic
• Non-stationary
  • Linear
  • Polynomial
• Non-Euclidean
  • Graph kernels

In addition to this, we implement kernel approximations using the Random Fourier feature approach.

Example

The following code snippet demonstrates how the first order Matérn kernel can be computed and, subsequently, approximated using random Fourier features.

import jaxkern as jk
import jax.numpy as jnp
import jax.random as jr
key = jr.PRNGKey(123)

# Define the points on which we'll evaluate the kernel
X = jr.uniform(key, shape = (10, 1), minval=-3., maxval=3.)
Y = jr.uniform(key, shape = (20, 1), minval=-3., maxval=3.)

# Instantiate the kernel and its parameters
kernel = jk.Matern32()
params = kernel.init_params(key)

# Compute the 10x10 Gram matrix
Kxx = kernel.gram(params, X)

# Compute the 10x20 cross-covariance matrix
Kxy = kernel.cross_covariance(params, X, Y)

# Build a RFF approximation
approx = RFF(kernel, num_basis_fns = 5)
rff_params = approx.init_params(key)

# Build an approximation to the Gram matrix
Qff = approx.gram(rff_params, X)

Code Structure

All kernels are supplied with a gram and cross_covariance method. When computing a Gram matrix, there is often some structure in the data (e.g., Markov) that can be exploited to yield a sparse matrix. To instruct JAX how to operate on this, the return type of gram is a Linear Operator from JaxLinOp.

Within GPJax, all kernel computations are handled using JaxKern.

Documentation

A full set of documentation is a work in progress. However, many of the details in JaxKern can be found in the GPJax documentation.