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- W74498904 abstract "The problem of finding a vector with the fewest nonzero elements that satisfies an underdetermined system of linear equations is an NP-complete problem that is typically solved numerically via convex heuristics or nicely-behaved non-convex relaxations. In this work we consider elementary methods based on projections for solving a sparse feasibility problem without employing convex heuristics. In a recent paper Bauschke, Luke, Phan and Wang (2013) showed that the fundamental method of alternating projections must converge locally linearly to a solution to the sparse feasibility problem with an affine constraint. Using different tools we also show local linear convergence of MAP without any assumptions on the problem structure. Additionally, we show that stronger assumptions than the sufficient conditions for linear convergence of MAP are in fact necessary for mere convergence of another prominent algorithm, the Douglas-Rachford algorithm. These necessary assumptions are not satisfied for most sparsity problems of interest. Still, we recover positive results locally for the Douglas-Rachford algorithm in some instances of the sparse feasibility problem with respect to the shadows of the iterates." @default.
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- W74498904 date "2013-07-08" @default.
- W74498904 modified "2023-09-27" @default.
- W74498904 title "Projection Methods for Sparse Affine Feasibility: results and counterexamples" @default.
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