Poisson reduced-rank models with sparse loadings

Abstract

High-dimensional Poisson reduced-rank models have been considered for statistical inference on low-dimensional locations of the individuals based on the observations of high-dimensional count vectors. In this study, we assume sparsity on a so-called loading matrix to enhance its interpretability. The sparsity assumption leads to the use of $L_1$ penalty, for the estimation of the loading. We provide novel computational and theoretical analyses for the corresponding penalized Poisson maximum likelihood estimation. We establish theoretical convergence rates for the parameters under weak-dependence conditions; this implies consistency even in large-dimensional problems. To implement the proposed method involving several computational issues, including nonconvex log-likelihoods, $L_1$ penalty, and orthogonal constraints, we developed an iterative algorithm. Further, we propose a Bayesian-Information-Criteria-based penalty parameter selection, which works well in the implementation. Some numerical evidence is provided by conducting real-data-based simulation analyses and the proposed method is illustrated with the analysis of German party manifesto data.