# A Matrix--free Likelihood Method for Exploratory Factor Analysis of High-dimensional Gaussian Data

## TL;DR

This paper introduces a fast, matrix-free likelihood method for high-dimensional Gaussian data in exploratory factor analysis, enabling efficient covariance estimation with fewer observations than variables.

## Contribution

It presents a novel profile likelihood approach combined with Lanczos and quasi-Newton algorithms for scalable, accurate covariance estimation in high-dimensional settings.

## Key findings

- Method is significantly faster than EM without losing accuracy.
- Successfully applied to real data on suicide attempters and controls.
- Demonstrates effectiveness in high-dimensional Gaussian datasets.

## Abstract

This paper proposes a novel profile likelihood method for estimating the covariance parameters in exploratory factor analysis of high-dimensional Gaussian datasets with fewer observations than number of variables. An implicitly restarted Lanczos algorithm and a limited-memory quasi-Newton method are implemented to develop a matrix-free framework for likelihood maximization. Simulation results show that our method is substantially faster than the expectation-maximization solution without sacrificing accuracy. Our method is applied to fit factor models on data from suicide attempters, suicide ideators and a control group.