Regularized Kaczmarz Algorithms for Tensor Recovery

TL;DR

This paper introduces a new stochastic Kaczmarz-based algorithmic framework for tensor recovery, enabling efficient real-time processing in applications like medical imaging and remote sensing.

Contribution

It develops a novel Kaczmarz algorithm framework for tensor recovery, with comprehensive convergence analysis and broad application demonstrations.

Findings

Effective in sparse signal recovery

Achieves low-rank tensor recovery

Performs well in image inpainting and deconvolution

Abstract

Tensor recovery has recently arisen in a lot of application fields, such as transportation, medical imaging and remote sensing. Under the assumption that signals possess sparse and/or low-rank structures, many tensor recovery methods have been developed to apply various regularization techniques together with the operator-splitting type of algorithms. Due to the unprecedented growth of data, it becomes increasingly desirable to use streamlined algorithms to achieve real-time computation, such as stochastic optimization algorithms that have recently emerged as an efficient family of methods in machine learning. In this work, we propose a novel algorithmic framework based on the Kaczmarz algorithm for tensor recovery. We provide thorough convergence analysis and its applications from the vector case to the tensor one. Numerical results on a variety of tensor recovery applications, including sparse signal recovery, low-rank tensor recovery, image inpainting and deconvolution, illustrate the enormous potential of the proposed methods.