Learning Markov Processes as Sum-of-Square Forms for Analytical Belief Propagation

TL;DR

This paper introduces a novel Sum-of-Squares based framework for analytical belief propagation in Markov process models, enabling efficient and scalable density estimation.

Contribution

It proposes a new functional form and training method for SoS models that preserve analytical belief propagation and scale to higher dimensions.

Findings

Achieves accuracy comparable to state-of-the-art methods.

Requires significantly less memory in low-dimensional spaces.

Successfully scales to 12D systems where existing methods fail beyond 2D.

Abstract

Harnessing the predictive capability of Markov process models requires propagating probability density functions (beliefs) through the model. For many existing models however, belief propagation is analytically infeasible, requiring approximation or sampling to generate predictions. This paper proposes a functional modeling framework leveraging sparse Sum-of-Squares (SoS) forms for valid (conditional) density estimation. We study the theoretical restrictions of modeling conditional densities using the SoS form, and propose a novel functional form for addressing such limitations. The proposed architecture enables generalized simultaneous learning of basis functions and coefficients, while preserving analytical belief propagation. In addition, we propose a training method that allows for exact adherence to the normalization and non-negativity constraints. Our results show that the proposed method achieves accuracy comparable to state-of-the-art approaches while requiring significantly less memory in low-dimensional spaces, and it further scales to 12D systems when existing methods fail beyond 2D.