A fast numerical method for max-convolution and the application to efficient max-product inference in Bayesian networks

TL;DR

This paper introduces a fast numerical method for max-convolution, enabling efficient max-product inference in Bayesian networks and related models, significantly reducing computational complexity for sums of discrete random variables.

Contribution

The paper presents a novel O(k log(k)) numerical approach for max-convolution, improving the efficiency of max-product inference in Bayesian networks and hidden Markov models.

Findings

Achieves O(k log(k)) complexity for max-convolution of nonnegative vectors.

Enables fast max-product inference on convolution trees with reduced runtime.

Potential applications include shortest path problems and HMMs with improved efficiency.

Abstract

Observations depending on sums of random variables are common throughout many fields; however, no efficient solution is currently known for performing max-product inference on these sums of general discrete distributions (max-product inference can be used to obtain maximum a posteriori estimates). The limiting step to max-product inference is the max-convolution problem (sometimes presented in log-transformed form and denoted as "infimal convolution", "min-convolution", or "convolution on the tropical semiring"), for which no O(k log(k)) method is currently known. Here I present a O(k log(k)) numerical method for estimating the max-convolution of two nonnegative vectors (e.g., two probability mass functions), where k is the length of the larger vector. This numerical max-convolution method is then demonstrated by performing fast max-product inference on a convolution tree, a data structure for performing fast inference given information on the sum of n discrete random variables in O(n k log(n k) log(n) ) steps (where each random variable has an arbitrary prior distribution on k contiguous possible states). The numerical max-convolution method can be applied to specialized classes of hidden Markov models to reduce the runtime of computing the Viterbi path from n k^2 to n k log(k), and has potential application to the all-pairs shortest paths problem.