Matrix Recovery with Implicitly Low-Rank Data

In this paper, we study the problem of matrix recovery, which aims to restore a target matrix of authentic samples from grossly corrupted observations. Most of the existing methods, such as the well-known Robust Principal Component Analysis (RPCA), assume that the target matrix we wish to recover is low-rank. However, the underlying data structure is often non-linear in practice, therefore the low-rankness assumption could be violated. To tackle this issue, we propose a novel method for matrix recovery in this paper, which could well handle the case where the target matrix is low-rank in an implicit feature space but high-rank or even full-rank in its original form. Namely, our method pursues the low-rank structure of the target matrix in an implicit feature space. By making use of the specifics of an accelerated proximal gradient based optimization algorithm, the proposed method could recover the target matrix with non-linear structures from its corrupted version. Comprehensive experiments on both synthetic and real datasets demonstrate the superiority of our method

Low-Rank Matrix Recovery from Noisy via an MDL Framework-based Atomic Norm

The recovery of the underlying low-rank structure of clean data corrupted with sparse noise/outliers is attracting increasing interest. However, in many low-level vision problems, the exact target rank of the underlying structure, the particular locations and values of the sparse outliers are not known. Thus, the conventional methods can not separate the low-rank and sparse components completely, especially gross outliers or deficient observations. Therefore, in this study, we employ the Minimum Description Length (MDL) principle and atomic norm for low-rank matrix recovery to overcome these limitations. First, we employ the atomic norm to find all the candidate atoms of low-rank and sparse terms, and then we minimize the description length of the model in order to select the appropriate atoms of low-rank and the sparse matrix, respectively. Our experimental analyses show that the proposed approach can obtain a higher success rate than the state-of-the-art methods even when the number of observations is limited or the corruption ratio is high. Experimental results about synthetic data and real sensing applications (high dynamic range imaging, background modeling, removing shadows and specularities) demonstrate the effectiveness, robustness and efficiency of the proposed method.Comment: 12 pages, 12 figure