Graphical Modelling without Independence Assumptions for Uncentered Data

TL;DR

This paper introduces a novel approach to multi-axis graphical modelling that relaxes the zero-mean assumption, enabling more accurate modeling of uncentered data through an efficient optimization method.

Contribution

It proposes the Kronecker-sum-structured mean assumption, allowing models to handle uncentered data without the zero-mean constraint, improving modeling accuracy.

Findings

The new assumption reduces modeling errors caused by zero-mean constraints.

Efficient coordinate descent algorithms solve the resulting nonconvex log-likelihoods.

The approach extends multi-axis graphical models to uncentered data.

Abstract

The independence assumption is a useful tool to increase the tractability of one's modelling framework. However, this assumption does not match reality; failing to take dependencies into account can cause models to fail dramatically. The field of multi-axis graphical modelling (also called multi-way modelling, Kronecker-separable modelling) has seen growth over the past decade, but these models require that the data have zero mean. In the multi-axis case, inference is typically done in the single sample scenario, making mean inference impossible. In this paper, we demonstrate how the zero-mean assumption can cause egregious modelling errors, as well as propose a relaxation to the zero-mean assumption that allows the avoidance of such errors. Specifically, we propose the "Kronecker-sum-structured mean" assumption, which leads to models with nonconvex-but-unimodal log-likelihoods that can be solved efficiently with coordinate descent.