Polynomial Threshold Functions of Bounded Tree-Width: Some Explainability and Complexity Aspects

TL;DR

This paper explores Boolean functions represented by polynomial threshold functions of bounded tree-width, focusing on their explainability in Bayesian classifiers and their computational complexity, with new insights into their representational power.

Contribution

It introduces the concept of polynomial threshold functions of bounded tree-width for Boolean functions and demonstrates their applications in explainable AI and Bayesian network classifiers.

Findings

Polynomial threshold functions of bounded tree-width can be used for explainable Bayesian classifiers.

A separation result shows differences in representational power between positive and general polynomial threshold functions.

Applications demonstrate the practical relevance of these functions in AI explainability.

Abstract

The tree-width of a multivariate polynomial is the tree-width of the hypergraph with hyperedges corresponding to its terms. Multivariate polynomials of bounded tree-width have been studied by Makowsky and Meer as a new sparsity condition that allows for polynomial solvability of problems which are intractable in general. We consider a variation on this theme for Boolean variables. A representation of a Boolean function as the sign of a polynomial is called a polynomial threshold representation. We discuss Boolean functions representable as polynomial threshold functions of bounded tree-width and present two applications to Bayesian network classifiers, a probabilistic graphical model. Both applications are in Explainable Artificial Intelligence (XAI), the research area dealing with the black-box nature of many recent machine learning models. We also give a separation result between the representational power of positive and general polynomial threshold functions.