  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.
