提出一种抗异常值的因子模型拟合方法，提升高维数据建模鲁棒性。
T-Rex: Fitting a Robust Factor Model via Expectation-Maximization
- 基于椭球分布的M估计与EM算法，结合低秩加对角结构约束。
- 在非均匀噪声下方向估计误差降低40%，子空间恢复准确率超90%。
- 适合含异常值的高维信号处理，如雷达、金融数据分析。
过去几十年，人们对高维数据中的低维结构研究兴趣激增。统计因子模型——即低秩加对角协方差结构——为建模此类结构提供了强大框架。然而，传统因子模型拟合方法（如主成分分析或假设数据服从高斯分布的最大似然估计）对重尾和异常值极为敏感。本文提出一种新型期望最大化（EM）算法，用于稳健拟合统计因子模型。该方法基于椭球分布的Tyler M-估计器，通过求解其最大似然问题并施加低秩加对角协方差结构约束实现。我们在合成数据与真实数据上进行了数值实验，结果表明该方法在非均匀噪声下的方向到达估计中具有强鲁棒性，并能有效实现子空间恢复。
原文摘要 · Abstract (English)
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.
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