Robustness of Sparse Recovery via $F$-minimization: A Topological ViewpointReport as inadecuate



 Robustness of Sparse Recovery via $F$-minimization: A Topological Viewpoint


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A recent trend in compressed sensing is to consider non-convex optimization techniques for sparse recovery. A general class of such optimizations, called $F$-minimization, has become of particular interest, since its exact reconstruction condition ERC in the noiseless setting can be precisely characterized by null space property NSP. However, little work has been done concerning its robust reconstruction condition RRC in the noisy setting. In this paper we look at the null space of the measurement matrix as a point on the Grassmann manifold, and then study the relation of the ERC and RRC sets on the Grassmannian. It is shown that the RRC set is exactly the topological interior of the ERC set. From this characterization, a previous result of the equivalence of ERC and RRC for $l p$-minimization follows easily as a special case. Moreover, when $F$ is non-decreasing, it is shown that the ERC and RRC sets are equivalent up to a set of measure zero. As a consequence, the probabilities of ERC and RRC are the same if the measurement matrix is randomly generated according to a continuous distribution. Finally, we provide several rules for comparing the performances of different cost functions, as applications of the above results.



Author: Jingbo Liu; Jian Jin; Yuantao Gu

Source: https://archive.org/







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