# Fitting Multilevel Factor Models

arXiv: https://arxiv.org/abs/2409.12067 · HTML: https://mlsanity.com/paper/2409.12067/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 3 of its 60 references that are in the map.
- First posted on arXiv 18 Sep 2024.

## Builds on (most cited first, 3 of 3)

- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [Covariance Prediction via Convex Optimization](https://mlsanity.com/paper/2101.12416/) (2021)
- [Factor Fitting, Rank Allocation, and Partitioning in Multilevel Low Rank Matrices](https://mlsanity.com/paper/2310.19214/) (2023)

## Built on by (most cited first, 1 of 1)

- [Customizing the Inductive Biases of Softmax Attention using Structured Matrices](https://mlsanity.com/paper/2509.07963/) (2025)

## Abstract (arXiv)

We examine a special case of the multilevel factor model, with covariance given by multilevel low rank (MLR) matrix~citeparshakova2023factor. 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.