Recovery of Coherent Data via Low-Rank Dictionary Pursuit

TL;DR

This paper addresses the limitations of RPCA in recovering low-rank matrices with clustered, coherent data by leveraging Low-Rank Representation with appropriately configured low-rank dictionaries, improving robustness.

Contribution

It introduces a theoretical framework showing low-rank dictionaries in LRR mitigate coherence issues, and proposes an unsupervised algorithm for dictionary learning.

Findings

LRR with low-rank dictionaries is immune to data coherence.

Proper dictionary configuration enhances low-rank matrix recovery.

Experimental results confirm the effectiveness of the proposed method.

Abstract

The recently established RPCA method provides us a convenient way to restore low-rank matrices from grossly corrupted observations. While elegant in theory and powerful in reality, RPCA may be not an ultimate solution to the low-rank matrix recovery problem. Indeed, its performance may not be perfect even when data are strictly low-rank. This is because conventional RPCA ignores the clustering structures of the data which are ubiquitous in modern applications. As the number of cluster grows, the coherence of data keeps increasing, and accordingly, the recovery performance of RPCA degrades. We show that the challenges raised by coherent data (i.e., the data with high coherence) could be alleviated by Low-Rank Representation (LRR), provided that the dictionary in LRR is configured appropriately. More precisely, we mathematically prove that if the dictionary itself is low-rank then LRR is immune to the coherence parameter which increases with the underlying cluster number. This provides an elementary principle for dealing with coherent data. Subsequently, we devise a practical algorithm to obtain proper dictionaries in unsupervised environments. Our extensive experiments on randomly generated matrices verify our claims.