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Upper-bounding $\ell_1$-optimization weak thresholds

In our recent work \cite{StojnicCSetam09} we considered solving under-determined systems of linear equations with sparse solutions. In a large dimensional and statistical context we proved that if the number of equations in the system is proportional to the length of the unknown vector then there is a sparsity (number of non-zero elements of the unknown vector) also proportional to the length of the unknown vector such that a polynomial $\ell_1$-optimization technique succeeds in solving the system. We provided lower bounds on the proportionality constants that are in a solid numerical agreement with what one can observe through numerical experiments. Here we create a mechanism that can be used to derive the upper bounds on the proportionality constants. Moreover, the upper bounds obtained through such a mechanism match the lower bounds from \cite{StojnicCSetam09} and ultimately make the latter ones optimal.