Stochastic expansions using continuous dictionaries: L\'{e}vy adaptive regression kernels

TL;DR

This paper introduces a novel nonparametric function estimation method using Lévy random fields to create flexible, continuous kernel expansions that adapt to local features, with proven convergence and Bayesian inference via MCMC.

Contribution

It develops a new class of priors for nonparametric estimation based on Lévy stochastic integrals, enabling adaptive, local feature modeling with convergence guarantees.

Findings

LARK method outperforms wavelet-based methods on test functions.

Provides convergence conditions in Besov and Sobolev norms.

Demonstrates flexibility in nonstationary applications.

Abstract

This article describes a new class of prior distributions for nonparametric function estimation. The unknown function is modeled as a limit of weighted sums of kernels or generator functions indexed by continuous parameters that control local and global features such as their translation, dilation, modulation and shape. L\'{e}vy random fields and their stochastic integrals are employed to induce prior distributions for the unknown functions or, equivalently, for the number of kernels and for the parameters governing their features. Scaling, shape, and other features of the generating functions are location-specific to allow quite different function properties in different parts of the space, as with wavelet bases and other methods employing overcomplete dictionaries. We provide conditions under which the stochastic expansions converge in specified Besov or Sobolev norms. Under a Gaussian error model, this may be viewed as a sparse regression problem, with regularization induced via the L\'{e}vy random field prior distribution. Posterior inference for the unknown functions is based on a reversible jump Markov chain Monte Carlo algorithm. We compare the L\'{e}vy Adaptive Regression Kernel (LARK) method to wavelet-based methods using some of the standard test functions, and illustrate its flexibility and adaptability in nonstationary applications.