Imaging geometry through dynamics: the observable representation

TL;DR

This paper introduces a method to recover the geometric structure of the underlying space of stochastic processes by analyzing the eigenvectors of transition probability matrices, revealing observable features of the process.

Contribution

It proposes a novel approach to infer the geometry of the state space from the spectral properties of transition matrices, focusing on slow eigenvectors as observables.

Findings

Eigenvectors with eigenvalues close to stationary eigenvector reveal geometric features.

The method enables reconstruction of the underlying coordinate space from stochastic data.

Observable eigenvectors provide insights into the structure of complex stochastic systems.

Abstract

For many stochastic processes there is an underlying coordinate space, $V$, with the process moving from point to point in $V$ or on variables (such as spin configurations) defined with respect to $V$. There is a matrix of transition probabilities (whether between points in $V$ or between variables defined on $V$) and we focus on its ``slow'' eigenvectors, those with eigenvalues closest to that of the stationary eigenvector. These eigenvectors are the ``observables,'' and they can be used to recover geometrical features of $V$.