Robust Tensor-on-Tensor Regression

Tensor-on-tensor (TOT) regression is an important tool for the analysis of tensor data, aiming to predict a set of response tensors from a corresponding set of predictor tensors. However, standard TOT regression is sensitive to outliers, which may be present in both the response and the predictor. It can be affected by casewise outliers, which are observations that deviate from the bulk of the data, as well as by cellwise outliers, which are individual anomalous cells within the tensors. The latter are particularly common due to the typically large number of cells in tensor data. This paper introduces a novel robust TOT regression method, named ROTOT, that can handle both types of outliers simultaneously, and can cope with missing values as well. This method uses a single loss function to reduce the influence of both casewise and cellwise outliers in the response. The outliers in the predictor are handled using a robust Multilinear Principal Component Analysis method. Graphical diagnostic tools are also proposed to identify the different types of outliers detected. The performance of ROTOT is evaluated through extensive simulations and further illustrated using the Labeled Faces in the Wild dataset, where ROTOT is applied to predict facial attributes.

Cellwise and Casewise Robust Covariance in High Dimensions

The sample covariance matrix is a cornerstone of multivariate statistics, but it is highly sensitive to outliers. These can be casewise outliers, such as cases belonging to a different population, or cellwise outliers, which are deviating cells (entries) of the data matrix. Recently some robust covariance estimators have been developed that can handle both types of outliers, but their computation is only feasible up to at most 20 dimensions. To remedy this we propose the cellRCov method, a robust covariance estimator that simultaneously handles casewise outliers, cellwise outliers, and missing data. It relies on a decomposition of the covariance on principal and orthogonal subspaces, leveraging recent work on robust PCA. It also employs a ridge-type regularization to stabilize the estimated covariance matrix. We establish some theoretical properties of cellRCov, including its casewise and cellwise influence functions as well as consistency and asymptotic normality. A simulation study demonstrates the superior performance of cellRCov in contaminated and missing data scenarios. Furthermore, its practical utility is illustrated in a real-world application to anomaly detection. We also construct and illustrate the cellRCCA method for robust and regularized canonical correlation analysis.

Robust Multilinear Principal Component Analysis

Multilinear Principal Component Analysis (MPCA) is an important tool for analyzing tensor data. It performs dimension reduction similar to PCA for multivariate data. However, standard MPCA is sensitive to outliers. It is highly influenced by observations deviating from the bulk of the data, called casewise outliers, as well as by individual outlying cells in the tensors, so-called cellwise outliers. This latter type of outlier is highly likely to occur in tensor data, as tensors typically consist of many cells. This paper introduces a novel robust MPCA method that can handle both types of outliers simultaneously, and can cope with missing values as well. This method uses a single loss function to reduce the influence of both casewise and cellwise outliers. The solution that minimizes this loss function is computed using an iteratively reweighted least squares algorithm with a robust initialization. Graphical diagnostic tools are also proposed to identify the different types of outliers that have been found by the new robust MPCA method. The performance of the method and associated graphical displays is assessed through simulations and illustrated on two real datasets.