Exact Decomposition of Joint Low Rankness and Local Smoothness Plus Sparse Matrices

TL;DR

This paper introduces a novel RPCA model that exactly decomposes matrices into low-rank, local smoothness, and sparse components, with theoretical guarantees and efficient algorithms, validated through experiments.

Contribution

The paper proposes the first theoretical guarantee for exact decomposition of matrices into low-rank, local smoothness, and sparse components using 3D correlated total variation regularization.

Findings

Exact decomposition achieved under mild assumptions.

Efficient ADMM algorithm with convergence guarantee.

Validated effectiveness on simulations and real data.

Abstract

It is known that the decomposition in low-rank and sparse matrices (\textbf{L+S} for short) can be achieved by several Robust PCA techniques. Besides the low rankness, the local smoothness (\textbf{LSS}) is a vitally essential prior for many real-world matrix data such as hyperspectral images and surveillance videos, which makes such matrices have low-rankness and local smoothness properties at the same time. This poses an interesting question: Can we make a matrix decomposition in terms of \textbf{L\&LSS +S } form exactly? To address this issue, we propose in this paper a new RPCA model based on three-dimensional correlated total variation regularization (3DCTV-RPCA for short) by fully exploiting and encoding the prior expression underlying such joint low-rank and local smoothness matrices. Specifically, using a modification of Golfing scheme, we prove that under some mild assumptions, the proposed 3DCTV-RPCA model can decompose both components exactly, which should be the first theoretical guarantee among all such related methods combining low rankness and local smoothness. In addition, by utilizing Fast Fourier Transform (FFT), we propose an efficient ADMM algorithm with a solid convergence guarantee for solving the resulting optimization problem. Finally, a series of experiments on both simulations and real applications are carried out to demonstrate the general validity of the proposed 3DCTV-RPCA model.