Interpreting Latent Variables in Factor Models via Convex Optimization

TL;DR

This paper introduces a convex optimization-based method to interpret latent variables in factor models by associating them with auxiliary covariates, providing theoretical guarantees and practical demonstrations.

Contribution

It presents a systematic, convex optimization approach to attribute semantic meaning to latent variables in factor analysis, extending existing methods with theoretical and empirical validation.

Findings

The method is computationally tractable and scalable.

It achieves consistent identification of latent-covariate associations.

Experimental results demonstrate practical utility on real data.

Abstract

Latent or unobserved phenomena pose a significant difficulty in data analysis as they induce complicated and confounding dependencies among a collection of observed variables. Factor analysis is a prominent multivariate statistical modeling approach that addresses this challenge by identifying the effects of (a small number of) latent variables on a set of observed variables. However, the latent variables in a factor model are purely mathematical objects that are derived from the observed phenomena, and they do not have any interpretation associated to them. A natural approach for attributing semantic information to the latent variables in a factor model is to obtain measurements of some additional plausibly useful covariates that may be related to the original set of observed variables, and to associate these auxiliary covariates to the latent variables. In this paper, we describe a systematic approach for identifying such associations. Our method is based on solving computationally tractable convex optimization problems, and it can be viewed as a generalization of the minimum-trace factor analysis procedure for fitting factor models via convex optimization. We analyze the theoretical consistency of our approach in a high-dimensional setting as well as its utility in practice via experimental demonstrations with real data.