Power of $\ell_1$-Norm Regularized Kaczmarz Algorithms for High-Order Tensor Recovery

TL;DR

This paper introduces novel $ ext{l}_1$-norm regularized Kaczmarz algorithms tailored for high-order tensor recovery, effectively exploiting sparsity and low-rank structures to improve image and video processing tasks.

Contribution

It proposes new Kaczmarz algorithms with $ ext{l}_1$-norm regularization for tensor reconstruction, including block and accelerated variants with convergence analysis.

Findings

Effective in image sequence destriping

Improves video deconvolution performance

Demonstrates strong results on synthetic and real data

Abstract

Tensors serve as a crucial tool in the representation and analysis of complex, multi-dimensional data. As data volumes continue to expand, there is an increasing demand for developing optimization algorithms that can directly operate on tensors to deliver fast and effective computations. Many problems in real-world applications can be formulated as the task of recovering high-order tensors characterized by sparse and/or low-rank structures. In this work, we propose novel Kaczmarz algorithms with a power of the $\ell_1$-norm regularization for reconstructing high-order tensors by exploiting sparsity and/or low-rankness of tensor data. In addition, we develop both a block and an accelerated variant, along with a thorough convergence analysis of these algorithms. A variety of numerical experiments on both synthetic and real-world datasets demonstrate the effectiveness and significant potential of the proposed methods in image and video processing tasks, such as image sequence destriping and video deconvolution.