Conditional probability tensor decompositions for multivariate categorical response regression

TL;DR

This paper introduces a novel low-rank tensor decomposition method for modeling multivariate categorical responses in regression, enabling scalable inference and capturing response dependencies effectively.

Contribution

It proposes a functional probability tensor decomposition approach that models complex response dependencies and can be efficiently fitted using a penalized EM algorithm.

Findings

Method performs well in simulations

Effective in modeling gene functional classes

Captures response dependencies accurately

Abstract

In many modern regression applications, the response consists of multiple categorical random variables whose probability mass is a function of a common set of predictors. In this article, we propose a new method for modeling such a probability mass function in settings where the number of response variables, the number of categories per response, and the dimension of the predictor are large. Our method relies on a functional probability tensor decomposition: a decomposition of a tensor-valued function such that its range is a restricted set of low-rank probability tensors. This decomposition is motivated by the connection between the conditional independence of responses, or lack thereof, and their probability tensor rank. We show that the model implied by such a low-rank functional probability tensor decomposition can be interpreted in terms of a mixture of regressions and can thus be fit using maximum likelihood. We derive an efficient and scalable penalized expectation maximization algorithm to fit this model and examine its statistical properties. We demonstrate the encouraging performance of our method through both simulation studies and an application to modeling the functional classes of genes.