Coherent set identification via direct low rank maximum likelihood estimation

TL;DR

This paper explores the relationship between two low rank modeling approaches for dynamical data, connecting coherence analysis with stochastic matrix factorization, and provides bounds and insights into their optimization objectives.

Contribution

It establishes a theoretical link between coherence detection and low rank stochastic matrix factorization, introducing bounds and new perspectives on their optimization.

Findings

DBMR results in a low rank projection of the full model

Bounds are derived between the objectives of the two approaches

Links are established between likelihood-based and projection-based methods

Abstract

We analyze connections between two low rank modeling approaches from the last decade for treating dynamical data. The first one is the coherence problem (or coherent set approach), where groups of states are sought that evolve under the action of a stochastic transition matrix in a way maximally distinguishable from other groups. The second one is a low rank factorization approach for stochastic matrices, called Direct Bayesian Model Reduction (DBMR), which estimates the low rank factors directly from observed data. We show that DBMR results in a low rank model that is a projection of the full model, and exploit this insight to infer bounds on a quantitative measure of coherence within the reduced model. Both approaches can be formulated as optimization problems, and we also prove a bound between their respective objectives. On a broader scope, this work relates the two classical loss functions of nonnegative matrix factorization, namely the Frobenius norm and the generalized Kullback--Leibler divergence, and suggests new links between likelihood-based and projection-based estimation of probabilistic models.