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Abstract: Recently, \cite{CRT,DonohoPol} theoretically analyzed the success of apolynomial $\ell 1$-optimization algorithm in solving an under-determinedsystem of linear equations. In a large dimensional and statistical context\cite{CRT,DonohoPol} proved that if the number of equations measurements inthe compressed sensing terminology in the system is proportional to the lengthof the unknown vector then there is a sparsity number of non-zero elements ofthe unknown vector also proportional to the length of the unknown vector suchthat $\ell 1$-optimization succeeds in solving the system. In this paper, weprovide an alternative performance analysis of $\ell 1$-optimization and obtainthe proportionality constants that in certain cases match or improve on thebest currently known ones from \cite{DonohoPol,DT}.



Author: Mihailo Stojnic

Source: https://arxiv.org/



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