Robust Covariance Estimation and Explainable Outlier Detection for Matrix-Valued Data

This work introduces the Matrix Minimum Covariance Determinant (MMCD) method, a novel robust location and covariance estimation procedure designed for data that are naturally represented in the form of a matrix. Unlike standard robust multivariate estimators, which would only be applicable after a vectorization of the matrix-variate samples leading to high-dimensional datasets, the MMCD estimators account for the matrix-variate data structure and consistently estimate the mean matrix, as well as the rowwise and columnwise covariance matrices in the class of matrix-variate elliptical distributions. Additionally, we show that the MMCD estimators are matrix affine equivariant and achieve a higher breakdown point than the maximal achievable one by any multivariate, affine equivariant location/covariance estimator when applied to the vectorized data. An efficient algorithm with convergence guarantees is proposed and implemented. As a result, robust Mahalanobis distances based on MMCD estimators offer a reliable tool for outlier detection. Additionally, we extend the concept of Shapley values for outlier explanation to the matrix-variate setting, enabling the decomposition of the squared Mahalanobis distances into contributions of the rows, columns, or individual cells of matrix-valued observations. Notably, both the theoretical guarantees and simulations show that the MMCD estimators outperform robust estimators based on vectorized observations, offering better computational efficiency and improved robustness. Moreover, real-world data examples demonstrate the practical relevance of the MMCD estimators and the resulting robust Shapley values.

本研究提出矩阵最小协方差行列式（Matrix Minimum Covariance Determinant, MMCD）方法，这是一种新颖的鲁棒位置与协方差估计流程，专为天然以矩阵形式表示的数据设计。与标准鲁棒多元估计器不同——后者仅在矩阵变量样本向量化后（这会导致高维数据集）才可应用——MMCD估计器考虑矩阵变量数据结构，并在矩阵变量椭圆分布类中一致估计均值矩阵，以及行向和列向协方差矩阵。此外，我们证明MMCD估计器具有矩阵仿射等变性（matrix affine equivariant），且其崩溃点（breakdown point）高于任何多元仿射等变位置/协方差估计器应用于向量化数据时所能达到的最大崩溃点。本文提出并实现了一种具有收敛保证的高效算法。因此，基于MMCD估计器的鲁棒马氏距离（Mahalanobis distances）为异常值检测提供了可靠工具。此外，我们将用于异常值解释的沙普利值（Shapley values）概念扩展到矩阵变量场景，能够将平方马氏距离分解为矩阵值观测的行、列或单个单元格的贡献。值得注意的是，理论保证与仿真结果均表明，MMCD估计器优于基于向量化观测的鲁棒估计器，具有更高的计算效率和更强的鲁棒性。此外，真实世界数据实例证明了MMCD估计器及所得鲁棒沙普利值的实际相关性。