Low Rank Regularization: A review

Low Rank Regularization (LRR), in essence, involves introducing a low rank or approximately low rank assumption to target we aim to learn, which has achieved great success in many data analysis tasks. Over the last decade, much progress has been made in theories and applications. Nevertheless, the intersection between these two lines is rare. In order to construct a bridge between practical applications and theoretical studies, in this paper we provide a comprehensive survey for LRR. Specifically, we first review the recent advances in two issues that all LRR models are faced with: (1) rank-norm relaxation, which seeks to find a relaxation to replace the rank minimization problem; (2) model optimization, which seeks to use an efficient optimization algorithm to solve the relaxed LRR models. For the first issue, we provide a detailed summarization for various relaxation functions and conclude that the non-convex relaxations can alleviate the punishment bias problem compared with the convex relaxations. For the second issue, we summarize the representative optimization algorithms used in previous studies, and analyze their advantages and disadvantages. As the main goal of this paper is to promote the application of non-convex relaxations, we conduct extensive experiments to compare different relaxation functions. The experimental results demonstrate that the non-convex relaxations generally provide a large advantage over the convex relaxations. Such a result is inspiring for further improving the performance of existing LRR models.

Brain Lesion Segmentation Based on Joint Constraints of Low-Rank Representation and Sparse Representation

The segmentation of brain lesions from a brain magnetic resonance (MR) image is of great significance for the clinical diagnosis and follow-up treatment. An automatic segmentation method for brain lesions is proposed based on the low-rank representation (LRR) and the sparse representation (SR) theory. The proposed method decomposes the brain image into the background part composed of brain tissue and the brain lesion part. Considering that each pixel in the brain tissue can be represented by the background dictionary, a low-rank representation that incorporates sparsity-inducing regularization term is adopted to model the part. Then, the linearized alternating direction method with adaptive penalty (LADMAP) was selected to solve the model, and the brain lesions can be obtained by the response of the residual matrix. The presented model not only reflects the global structure of the image but also preserves the local information of the pixels, thus improving the representation accuracy. The experimental results on the data of brain tumor patients and multiple sclerosis patients revealed that the proposed method is superior to several existing methods in terms of segmentation accuracy while realizing the segmentation automatically.