Poisson PCA for matrix count data

TL;DR

This paper introduces Poisson PCA, a dimension reduction method for matrix count data based on latent normal variables, with proven consistency and superior performance over existing methods.

Contribution

It develops a novel Poisson PCA model for count matrices, including parameter estimation, latent dimension selection, and a sparsity extension, outperforming existing approaches.

Findings

Method surpasses vectorization-based competitors.

Estimates model parameters with root-n consistency.

Performs well on simulated and real data.

Abstract

We develop a dimension reduction framework for data consisting of matrices of counts. Our model is based on assuming the existence of a small amount of independent normal latent variables that drive the dependency structure of the observed data, and can be seen as the exact discrete analogue for a contaminated low-rank matrix normal model. We derive estimators for the model parameters and establish their root-$n$ consistency. An extension of a recent proposal from the literature is used to estimate the latent dimension of the model. Additionally, a sparsity-accommodating variant of the model is considered. The method is shown to surpass both its vectorization-based competitors and matrix methods assuming the continuity of the data distribution in analysing simulated data and real abundance data.