Matrix-Product Belief Propagation for continuous-state-space variables
Abstract
Computation of observables in discrete stochastic, possibly conditioned, dynamics over large sparse networks is fundamental for a myriad of applications. The Matrix-Product Belief Propagation method allows a semi-analytical estimation of such observables in tree-like networks with a controlled error that depends on the bond dimension, that is the dimension of the employed square matrices. 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. For a wide class of models, the computational cost of the method is linear in the network size with a prefactor that depends on the basis size and the bond dimension. 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.
Related
BibTeX
@misc{florio2026matrixproduct,
title = {{Matrix-Product Belief Propagation for continuous-state-space variables}},
author = {Federico Florio and Alfredo Braunstein},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2605.14522},
eprint = {2605.14522},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.14522},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Matrix-Product Belief Propagation for continuous-state-space variables
AU - Federico Florio
AU - Alfredo Braunstein
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2605.14522
UR - https://arxiv.org/abs/2605.14522
AB - Computation of observables in discrete stochastic, possibly conditioned, dynamics over large sparse networks is fundamental for a myriad of applications. The Matrix-Product Belief Propagation method allows a semi-analytical estimation of such observables in tree-like networks with a controlled error that depends on the bond dimension, that is the dimension of the employed square matrices. 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. For a wide class of models, the computational cost of the method is linear in the network size with a prefactor that depends on the basis size and the bond dimension. 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.
ER -