Fast operator learning for mapping correlations

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.

💡 Research Summary

The paper introduces a fast, optimization‑free framework for learning the transition operator of high‑dimensional reversible Markov processes. The authors start by projecting the infinite‑dimensional transition operator (P_t) onto a finite set of basis functions ({\psi_i}{i=1}^{N_b}) using a Galerkin approach, defining the transition moment matrix (M) with entries (M{ij}= \langle \psi_i, P_t\psi_j\rangle_\mu). Two choices of the reference measure (\mu) are considered: a separable mean‑field density and the equilibrium density of the process. Under either measure, (P_t) and its generator are self‑adjoint and positive semi‑definite, guaranteeing that (M) inherits these properties.

To avoid the exponential blow‑up associated with tensor‑product bases, the authors employ a two‑cluster basis: for each pair of dimensions ((i_1,i_2)) they take the product of one‑dimensional orthonormal functions (\phi_{i_1j_1}) and (\phi_{i_2j_2}). The total number of basis functions is ((dn)^2) (with (n) one‑dimensional functions per coordinate), which scales quadratically rather than exponentially with the ambient dimension (d). This basis is well‑suited to capture pairwise correlations that dominate many‑body physical systems.

Each entry of (M) involves a high‑dimensional expectation. The authors estimate it by Monte‑Carlo: draw (N_{\text{src}}) independent samples from (\mu), and for each sample launch (N_{\text{traj}}) short trajectories of length (t). The empirical estimator \