Relation between exact and robust recovery for F-minimization: A topological viewpoint

Recent work in compressed sensing has shown the possibility reducing the number of measurements via non-convex optimization methods. Most of these methods can be studied in the general framework called “F-minimization”, for which the relation between the noiseless exact recovery condition (ERC) and noisy robust recovery condition (RRC) was not fully understood. In this paper, we associate each set of nulls spaces of the measurement matrices satisfying ERC/RRC as a subset of a Grassmannian, and show that the RRC set is exactly the interior of the ERC set. Then, a previous result of the equivalence of ERC and RRC for l_p-minimization follows easily as a special case. We also show under some mild but necessary additional assumption that the ERC and RRC sets differ by a set of measure zero.