Matrix-Product Belief Propagation for continuous-state-space variables

TL;DR

This paper extends the Matrix-Product Belief Propagation method to continuous and mixed variables using a Hilbert basis expansion, enabling efficient semi-analytical computations for complex stochastic network models.

Contribution

It introduces a generalized, scalable approach for continuous/discrete models, demonstrated on Kinetic Ising dynamics with real-valued couplings.

Findings

Accurate estimation of observables compared to Monte Carlo simulations.

Linear computational cost with network size, depending on basis and bond sizes.

Successful calculation of time correlations, energy, magnetization, and large deviation functions.

Abstract

Computation of observables in discrete stochastic, possibly conditioned, dynamics over large sparse networks is at the basis of a myriad of applications. The Matrix-Product Belief Propagation method allows a semi-analytical estimation of such observables with a controlled error that depends on the size of the employed matrices, called bond size. Its computational cost is linear in the time horizon and the network size for a large family of models with discrete degrees of freedom. Here, a generalization of this method to models with continuous or mixed continuous/discrete degrees of freedom is presented, using a tunable expansion in a Hilbert function basis. The computational cost of the method is linear in the network size with a prefactor that depends on the basis size and the bond size. The method's efficacy is demonstrated by employing a Fourier basis for a mixed continuous/discrete representation of the Kinetic Ising dynamics with real-valued random couplings, where intermediate ``local fields'' are treated as continuous. The accuracy of the method is verified via comparison with Monte-Carlo simulations. For this model, we calculate time auto-correlations, time evolution of energy and magnetization, and finally we estimate the large deviation function of the magnetization at a given future time.