Fast operator learning for mapping correlations

TL;DR

This paper introduces a fast, optimization-free approach for learning transition operators in high-dimensional Markov processes by leveraging low-rank structures and correlation decay, enabling efficient prediction and simulation.

Contribution

It presents a novel Galerkin projection method that captures correlations without curse of dimensionality, with theoretical error analysis and practical efficiency for high-dimensional problems.

Findings

Efficient prediction of future events in high dimensions.

Low-rank, compressed operator representations reduce computational complexity.

Method enables high-dimensional rare-events simulations.

Abstract

We propose a fast, optimization-free method for learning the transition operators of high-dimensional Markov processes. The central idea is to perform a Galerkin projection of the transition operator to a suitable set of low-order bases that capture the correlations between the dimensions. Such a discretized operator can be obtained from moments corresponding to our choice of basis without curse of dimensionality. Furthermore, by exploiting its low-rank structure and the spatial decay of correlations, we can obtain a compressed representation with computational complexity of order $\mathcal{O}(dN)$, where $d$ is the dimensionality and $N$ is the sample size. We further theoretically analyze the approximation error of the proposed compressed representation. We numerically demonstrate that the learned operator allows efficient prediction of future events and solving high-dimensional boundary value problems. This gives rise to a simple linear algebraic method for high-dimensional rare-events simulations.