Fitting Multilevel Factor Models

We examine a special case of the multilevel factor model, with covariance given by multilevel low rank (MLR) matrix [T. Parshakova et al., Factor fitting, rank allocation, and partitioning in multilevel low rank matrices, in Optimization, Discrete Mathematics, and Applications to Data Sciences, SOIA 220, Springer, 2024, pp. 135–173]. 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. Reproducibility of computational results. This paper has been awarded the “SIAM Reproducibility Badge: Code and data available” as a recognition that the authors have followed reproducibility principles valued by SIMAX and the scientific computing community. Code and data that allow readers to reproduce the results in this paper are available at https://github.com/cvxgrp/multilevel_factor_model (see notebooks in /examples folder). [Formula: see text]