Reduced Complexity Filtering with Stochastic Dominance Bounds: A Convex Optimization Approach

TL;DR

This paper introduces a convex optimization-based method to construct low-rank stochastic matrices that provide provable upper and lower bounds for Hidden Markov Model filters, significantly reducing computational complexity.

Contribution

It presents a novel convex optimization approach using nuclear norm minimization with copositive constraints to efficiently bound HMM filters with low-rank matrices.

Findings

Bounds are computationally efficient with O(XR) complexity.

Monte-Carlo importance sampling leverages bounds for posterior estimation.

Explicit bounds on the variational norm between true and bounded posteriors are derived.

Abstract

This paper uses stochastic dominance principles to construct upper and lower sample path bounds for Hidden Markov Model (HMM) filters. Given a HMM, by using convex optimization methods for nuclear norm minimization with copositive constraints, we construct low rank stochastic marices so that the optimal filters using these matrices provably lower and upper bound (with respect to a partially ordered set) the true filtered distribution at each time instant. Since these matrices are low rank (say R), the computational cost of evaluating the filtering bounds is O(XR) instead of O(X2). A Monte-Carlo importance sampling filter is presented that exploits these upper and lower bounds to estimate the optimal posterior. Finally, using the Dobrushin coefficient, explicit bounds are given on the variational norm between the true posterior and the upper and lower bounds.