Asymptotic Analysis of a Matrix Latent Decomposition Model

¹ ARAMIS Project Team, Inria, Paris 75013, France
² ARAMIS Lab, Brain and Spine Institute, ICM, INSERM UMR 1127, CNRS UMR 7225, Sorbonne Universite, Hopital de la Pitie-Salpetriere, Paris 75013, France
³ CMAP, Ecole polytechnique, Palaiseau 91120, France
⁴ Centre de Recherche des Cordeliers, Universite de Paris, INSERM UMR 1138, Sorbonne Universite, Paris 75006, France
⁵ HEKA Project Team, Inria, Paris 75006, France

^* Corresponding author: clement.mantoux@inria.fr

Received: 7 February 2022
Accepted: 7 April 2022

Abstract

Matrix data sets arise in network analysis for medical applications, where each network belongs to a subject and represents a measurable phenotype. These large dimensional data are often modeled using lower-dimensional latent variables, which explain most of the observed variability and can be used for predictive purposes. In this paper, we provide asymptotic convergence guarantees for the estimation of a hierarchical statistical model for matrix data sets. It captures the variability of matrices by modeling a truncation of their eigendecomposition. We show that this model is identifiable, and that consistent Maximum A Posteriori (MAP) estimation can be performed to estimate the distribution of eigenvalues and eigenvectors. The MAP estimator is shown to be asymptotically normal for a restricted version of the model.