Recovering Joint Probability of Discrete Random Variables from Pairwise Marginals

TL;DR

This paper introduces a new framework for recovering the joint probability mass function of discrete random variables using only pairwise marginals, reducing sample complexity and improving efficiency compared to previous methods that rely on higher-order marginals.

Contribution

The work proposes a coupled nonnegative matrix factorization approach and a Gram-Schmidt-like algorithm for joint PMF recovery from pairwise marginals, with theoretical guarantees and practical improvements.

Findings

The method achieves accurate joint PMF recovery with lower sample complexity.

The Gram-Schmidt-like algorithm provably converges with bounded error.

An EM algorithm further enhances accuracy and efficiency.

Abstract

Learning the joint probability of random variables (RVs) is the cornerstone of statistical signal processing and machine learning. However, direct nonparametric estimation for high-dimensional joint probability is in general impossible, due to the curse of dimensionality. Recent work has proposed to recover the joint probability mass function (PMF) of an arbitrary number of RVs from three-dimensional marginals, leveraging the algebraic properties of low-rank tensor decomposition and the (unknown) dependence among the RVs. Nonetheless, accurately estimating three-dimensional marginals can still be costly in terms of sample complexity, affecting the performance of this line of work in practice in the sample-starved regime. Using three-dimensional marginals also involves challenging tensor decomposition problems whose tractability is unclear. This work puts forth a new framework for learning the joint PMF using only pairwise marginals, which naturally enjoys a lower sample complexity relative to the third-order ones. A coupled nonnegative matrix factorization (CNMF) framework is developed, and its joint PMF recovery guarantees under various conditions are analyzed. Our method also features a Gram--Schmidt (GS)-like algorithm that exhibits competitive runtime performance. The algorithm is shown to provably recover the joint PMF up to bounded error in finite iterations, under reasonable conditions. It is also shown that a recently proposed economical expectation maximization (EM) algorithm guarantees to improve upon the GS-like algorithm's output, thereby further lifting up the accuracy and efficiency. Real-data experiments are employed to showcase the effectiveness.