Maximum likelihood inference for high-dimensional problems with multiaffine variable relations

TL;DR

This paper introduces AIRLS, a novel algorithm for maximum likelihood inference in high-dimensional models with multiaffine variable relations, demonstrating superior scalability and robustness over existing methods.

Contribution

The paper proposes AIRLS, a new convergent algorithm for high-dimensional multiaffine models, with an efficient variance computation and applications to graphical models.

Findings

AIRLS outperforms state-of-the-art methods in scalability and robustness.

The algorithm exhibits empirically observed super-linear convergence.

Numerical experiments confirm improved convergence speed and noise robustness.

Abstract

Maximum Likelihood Estimation of continuous variable models can be very challenging in high dimensions, due to potentially complex probability distributions. The existence of multiple interdependencies among variables can make it very difficult to establish convergence guarantees. This leads to a wide use of brute-force methods, such as grid searching and Monte-Carlo sampling and, when applicable, complex and problem-specific algorithms. In this paper, we consider inference problems where the variables are related by multiaffine expressions. We propose a novel Alternating and Iteratively-Reweighted Least Squares (AIRLS) algorithm, and prove its convergence for problems with Generalized Normal Distributions. We also provide an efficient method to compute the variance of the estimates obtained using AIRLS. Finally, we show how the method can be applied to graphical statistical models. We perform numerical experiments on several inference problems, showing significantly better performance than state-of-the-art approaches in terms of scalability, robustness to noise, and convergence speed due to an empirically observed super-linear convergence rate.