A two-way factor model for high-dimensional matrix data

TL;DR

This paper introduces a two-way factor model for high-dimensional matrix data, capturing row and column correlations with low-dimensional factors, and develops an estimation method with proven properties.

Contribution

It proposes a novel two-way factor model with separable row and column effects and provides an efficient estimation strategy with theoretical guarantees.

Findings

The MLE's variance depends on the variance distance between row and column factors.

The proposed method achieves consistent estimation as data dimensions grow.

Simulation and real data demonstrate the model's effectiveness.

Abstract

In this article, we introduce a two-way factor model for a high-dimensional data matrix and study the properties of the maximum likelihood estimation (MLE). The proposed model assumes separable effects of row and column attributes and captures the correlation across rows and columns with low-dimensional hidden factors. The model inherits the dimension-reduction feature of classical factor models but introduces a new framework with separable row and column factors, representing the covariance or correlation structure in the data matrix. We propose a block alternating, maximizing strategy to compute the MLE of factor loadings as well as other model parameters. We discuss model identifiability, obtain consistency and the asymptotic distribution for the MLE as the numbers of rows and columns in the data matrix increase. One interesting phenomenon that we learned from our analysis is that the variance of the estimates in the two-way factor model depends on the distance of variances of row factors and column factors in a way that is not expected in classical factor analysis. We further demonstrate the performance of the proposed method through simulation and real data analysis.