Upper-bounding ℓ1-optimization weak thresholds

In our recent work [49] we considered solving under-determi ned systems of linear equations with sparse solutions. In a large dimensional and statistical context w e proved that if the number of equations in the system is proportional to the length of the unknown vector th en there is a sparsity (number of non-zero elements of the unknown vector) also proportional to the length of t e unknown vector such that a polynomial l1-optimization technique succeeds in solving the system. We provided lower bounds on the proportionality constants that are in a solid numerical agreement with what o ne 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 mechanis m match the lower bounds from [49] and ultimately make the latter ones optimal.