Robust Low-rank Tensor Decomposition with the $L_2$ Criterion

The growing prevalence of tensor data, or multiway arrays, in science and engineering applications motivates the need for tensor decompositions that are robust against outliers. In this paper, we present a robust Tucker decomposition estimator based on the $L_2$ criterion, called the Tucker-$L_{2}E$. Our numerical experiments demonstrate that Tucker-$L_{2}E$ has empirically stronger recovery performance in more challenging high-rank scenarios compared with existing alternatives. The appropriate Tucker-rank can be selected in a data-driven manner with cross-validation or hold-out validation. The practical effectiveness of Tucker-$L_{2}E$ is validated on real data applications in fMRI tensor denoising, PARAFAC analysis of fluorescence data, and feature extraction for classification of corrupted images.