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

Karine Chubarian, Johnny Joyce, Gyorgy Turan

TL;DR

This paper investigates polynomial threshold representations (PTFs) of Boolean functions through the lens of tree-width, linking algebraic structure to tractability and explainability. It shows how Bayesian network classifiers of bounded tree-width map to PTFs with bounded width (TAN giving a quadratic QTF, general bounded-tree-width networks yielding path-width $O(k\log n)$) and explores approximate knowledge compilation by producing polynomial-size OBDDs that approximate classifiers under the input distribution. A concrete application to explainability is provided by viewing Bayesian network classifiers as generalized additive models with interactions (GA$^2$M) and evaluating explainability via InterpretML on a small TAN example, achieving meaningful term overlap with ground truth. The paper also establishes a separation result: there exists a monotone function with a linear-size QTF but quadratic-size positive QTFs, underscoring limits of positive representations. Overall, the work advances explainability and complexity analyses for bounded-tree-width PTFs and their use in Bayesian networks and XAI.

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.

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

TL;DR

This paper investigates polynomial threshold representations (PTFs) of Boolean functions through the lens of tree-width, linking algebraic structure to tractability and explainability. It shows how Bayesian network classifiers of bounded tree-width map to PTFs with bounded width (TAN giving a quadratic QTF, general bounded-tree-width networks yielding path-width ) and explores approximate knowledge compilation by producing polynomial-size OBDDs that approximate classifiers under the input distribution. A concrete application to explainability is provided by viewing Bayesian network classifiers as generalized additive models with interactions (GAM) and evaluating explainability via InterpretML on a small TAN example, achieving meaningful term overlap with ground truth. The paper also establishes a separation result: there exists a monotone function with a linear-size QTF but quadratic-size positive QTFs, underscoring limits of positive representations. Overall, the work advances explainability and complexity analyses for bounded-tree-width PTFs and their use in Bayesian networks and XAI.

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.
Paper Structure (12 sections, 8 theorems, 29 equations, 3 figures, 2 tables)

This paper contains 12 sections, 8 theorems, 29 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Boolean functions
  4. Widths
  5. Bayesian Network Classifiers
  6. Bayesian Network Classifiers and Polynomial Threshold Functions
  7. Approximating Bayesian Network Classifiers with OBDD
  8. Ordered Binary Decision Diagrams
  9. Result and Proof Outline
  10. Bayesian Network Classifiers for Evaluating the Explainability of Generalized Additive Models with Interactions
  11. Experimental Results on a Small Example
  12. A Complexity Lower Bound for Positive Polynomial Threshold Representations

Key Result

Proposition 1

Let $N$ be a Bayesian network classifier with non-zero conditional probabilities. Then there is a polynomial $p$ such that where $p$ is of degree at most $d_N$ and every term is a subset of a family. Thus $f_N$ is a PTF of degree at most $d_N$.

Figures (3)

  • Figure 1: OBDDs for function (\ref{['eq:ex']}).
  • Figure 2: The TAN used in the experiment.
  • Figure 3: Optimal classifier versus learned classifiers: accuracy and term overlap

Theorems & Definitions (12)

  • Proposition 1
  • proof
  • Corollary 2
  • Corollary 3
  • proof
  • Theorem 4
  • proof
  • Lemma 5
  • Lemma 6
  • Lemma 7
  • ...and 2 more