Chapter 5 - Generalized thresholding for low-rank tensor recovery: approaches based on model and learning

Abstract

Iterative shrinkage/thresholding algorithms have been widely used in sparse regression, compressive sensing, signal restoration, and data mining. As tensor can better preserve the multilinear structure in multiway data processing, tensor singular value thresholding is an effective way for low-rank tensor approximation. However, its performance is closely related to the choice of thresholding operators. In this chapter, we summarize a list of thresholding operators corresponding to various convex or nonconvex regularizations, which can be used for tensor singular value thresholding. Focusing on low-rank tensor recovery, we discuss generalized tensor singular value thresholding algorithms on the formulations, convergence, and penalty selection. In addition to traditional model-based thresholding algorithms, deep unrolling and deep plug-and-play methods are used as generalized thresholding algorithms in low-rank tensor recovery. Deep unrolling can transform iterative thresholding into deep networks, whilst deep plug-and-play takes a deep neural network to generalize the thresholding operator in tensor recovery. As two typical data processing techniques based on low-rank tensor approximation, we take tensor completion and robust tensor principal component analysis as examples to demonstrate the performance enhancement by the generalized thresholding methods.