Fast operator learning for mapping correlations

TL;DR

This work introduces an optimization-free framework to learn fast transition operators for high-dimensional Markov processes by projecting the transition operator onto a two-cluster basis, yielding a compact transition moment matrix. A two-level compression—global CUR-based low-rank factorization followed by a local sparse-plus-low-rank refinement—achieves memory $\mathcal{O}(d)$ and compute $\mathcal{O}(dN)$, leveraging decay of correlations to maintain accuracy. The authors provide theoretical guarantees under reversible finite-state assumptions, showing exponential decay of correlation and provable error bounds for the compressed representation. Numerical experiments on high-dimensional Langevin-type dynamics demonstrate efficient moment prediction, density prediction via adjoints, and committor solutions, highlighting the method’s practicality for rare-events analysis without optimization. Overall, the approach offers a scalable, mathematically grounded route to approximate high-dimensional transition operators and solve associated boundary-value problems.

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.

Figures (9)

• Figure 1: Two-level compression of transition moment matrix.
• Figure 2: Global minimizer $x_-$ (left) and $x_+$ (right) for $V(x)$.
• Figure 3: The leading 400 $\log_{10}$-scaled singular values of the full transition moment matrices evaluated at lag times $t=0.0001,0.01,1$, after normalizing each set of singular values by the largest singular value.
• Figure 4: $\log_{10}$-scaled correlation $\lvert \langle g_i, g_j \rangle_{\rho_\infty} - \langle g_i \rangle_{\rho_\infty} \langle g_j \rangle_{\rho_\infty} \rvert$ versus the site distance $|i-j|$ for one-cluster functions $g_i(x) = (\sin(x_i))^2$.
• Figure 5: (A) Matricization of a representative slice $M_{\rho_\infty}^t(:,j)$ of the transition moment matrix; (B) sparse component $P_j$; (C) low-rank residual $Q_j = M_{\rho_\infty}^t(:,j) - P_j$; (D) $\log_{10}$-scaled spectrum of the original slice $M_{\rho_\infty}^t(:,j)$ and the residual $Q_j$.

Theorems & Definitions (30)

• Proposition 2.1.1
• Remark 1
• Remark 2
• Remark 3
• Definition 6.1.1
• Lemma 6.2.1
• proof
• Lemma 6.2.2: Kressner, p. 23-25
• Lemma 6.2.3: Decay of correlation
• Lemma 6.2.4