Robust Low-Rank Modelling on Matrices and Tensors

Robust low-rank modelling has recently emerged as a family of powerful methods for recovering the low-dimensional structure of grossly corrupted data, and has become successful in a wide range of applications in signal processing and computer vision. In principle, robust low-rank modelling focuses on decomposing a given data matrix into a low-rank and a sparse component, by minimising the rank of the former and maximising the sparsity of the latter. In several practical cases, however, tensors can be a more suitable representation than matrices for higher-order data. Nevertheless, existing work on robust low-rank modelling has mostly focused on matrices and has largely neglected tensors. In this thesis we aim to bridge the gap between matrix and tensor methods for robust lowrank modelling. We conduct a thorough survey on matrix methods, in which we discuss and analyse the state-of-the-art algorithms. We take one step further and we present—for the first time in the literature—a set of novel algorithms for robust low-rank modelling on tensors. We show how both matrix and tensor methods can be extended to deal with missing values and we discuss non-convex generalisations thereof. Finally, we demonstrate the applicability of all methods, both those already existing and those proposed herein, to a range of computer vision problems of practical interest.