Fitting Multilevel Factor Models

TL;DR

This paper introduces a fast, scalable EM algorithm for multilevel factor models with MLR covariance, enabling efficient hierarchical data analysis with linear complexity and an open-source implementation.

Contribution

It develops a novel, efficient EM algorithm tailored for multilevel factor models with MLR covariance, including new techniques for matrix inversion and Cholesky factorization.

Findings

Linear time and storage complexity per iteration

Efficient inverse computation for MLR matrices

Open-source implementation of the methods

Abstract

We examine a special case of the multilevel factor model, with covariance given by multilevel low rank (MLR) matrix~\cite{parshakova2023factor}. We develop a novel, fast implementation of the expectation-maximization algorithm, tailored for multilevel factor models, to maximize the likelihood of the observed data. This method accommodates any hierarchical structure and maintains linear time and storage complexities per iteration. This is achieved through a new efficient technique for computing the inverse of the positive definite MLR matrix. We show that the inverse of positive definite MLR matrix is also an MLR matrix with the same sparsity in factors, and we use the recursive Sherman-Morrison-Woodbury matrix identity to obtain the factors of the inverse. Additionally, we present an algorithm that computes the Cholesky factorization of an expanded matrix with linear time and space complexities, yielding the covariance matrix as its Schur complement. This paper is accompanied by an open-source package that implements the proposed methods.