AFD Types Sparse Representations vs. the Karhunen-Loeve Expansion for Decomposing Stochastic Processes

TL;DR

This paper introduces adaptive Fourier decomposition (AFD) methods for stochastic processes, demonstrating they are computationally more efficient and better at capturing local details than the traditional Karhunen-Loeve (KL) expansion.

Contribution

The paper develops AFD-based algorithms for stochastic processes that match KL convergence rates but require less computation and better capture local features.

Findings

AFD methods have similar convergence rates to KL decomposition.

AFD reduces computational complexity by avoiding eigenvalue calculations.

AFD outperforms KL in describing local details of stochastic processes.

Abstract

This article introduces adaptive Fourier decomposition (AFD) type methods, emphasizing on those that can be applied to stochastic processes and random fields, mainly including stochastic adaptive Fourier decomposition and stochastic pre-orthogonal adaptive Fourier decomposition. We establish their algorithms based on the covariant function and prove that they enjoy the same convergence rate as the Karhunen-Lo\`eve (KL) decomposition. The AFD type methods are compared with the KL decomposition. In contrast with the latter, the AFD type methods do not need to compute eigenvalues and eigenfunctions of the kernel-integral operator induced by the covariance function, and thus considerably reduce the computation complexity and computer consumes. Various kinds of dictionaries offer AFD flexibility to solve problems of a great variety, including different types of deterministic and stochastic equations. The conducted experiments show, besides the numerical convenience and fast convergence, that the AFD type decompositions outperform the KL type in describing local details, in spite of the proven global optimality of the latter.