Sparse representation (L1 minimization) via half-quadratic minimization

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Ran He, Wei-Shi Zheng,Tieniu Tan, and Zhenan Sun. Half-quadratic based Iterative Minimization for Robust Sparse Representation. Submitted to IEEE Trans. on Pattern Analysis and Machine Intelligence.
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Robust sparse representation has shown significant potential in solving challenging problems in computer vision such as biometrics and visual surveillance. Although several robust sparse models have been proposed and promising results have been obtained, they are either for error detection or error correction, and learning a general framework that systematically unifies these two concepts is still an open problem. In this paper, we develop a half-quadratic (HQ) framework to solve robust sparse coding problems. By defining different kinds of half-quadratic functions, the proposed HQ framework is applicable to perform both error correction and error detection. More specifically, by using the additive form of HQ, we propose an L1-regularized error correction algorithm by iteratively recovering corrupted data from errors incurred by noise and outliers; by using the multiplicative form of HQ, we propose an L1-regularized error detection algorithm by learning from uncorrupted data iteratively.

These published codes are the algorithms in the appendix of our submitted TPAMI paper.
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