T-Rex: Fitting a Robust Factor Model via Expectation-Maximization
This addresses the sensitivity of traditional factor model methods to outliers, offering a robust alternative for applications in signal processing and data analysis.
The paper tackles the problem of fitting statistical factor models to high-dimensional data with heavy tails and outliers, proposing a robust expectation-maximization algorithm based on Tyler's M-estimator, and demonstrates its effectiveness in synthetic and real examples like direction-of-arrival estimation and subspace recovery.
Over the past decades, there has been a surge of interest in studying low-dimensional structures within high-dimensional data. Statistical factor models $-$ i.e., low-rank plus diagonal covariance structures $-$ offer a powerful framework for modeling such structures. However, traditional methods for fitting statistical factor models, such as principal component analysis (PCA) or maximum likelihood estimation assuming the data is Gaussian, are highly sensitive to heavy tails and outliers in the observed data. In this paper, we propose a novel expectation-maximization (EM) algorithm for robustly fitting statistical factor models. Our approach is based on Tyler's M-estimator of the scatter matrix for an elliptical distribution, and consists of solving Tyler's maximum likelihood estimation problem while imposing a structural constraint that enforces the low-rank plus diagonal covariance structure. We present numerical experiments on both synthetic and real examples, demonstrating the robustness of our method for direction-of-arrival estimation in nonuniform noise and subspace recovery.