Robust Tensor-on-Tensor Regression

TL;DR

This paper introduces ROTOT, a robust tensor-on-tensor regression method that effectively handles both casewise and cellwise outliers, missing data, and provides diagnostic tools, improving tensor data analysis.

Contribution

The paper proposes a novel robust TOT regression approach, ROTOT, capable of managing various outliers and missing data simultaneously, with diagnostic tools for outlier detection.

Findings

ROTOT outperforms existing methods in simulations.

ROTOT effectively detects different outlier types.

Application to facial attribute prediction demonstrates practical utility.

Abstract

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.