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The recent work of \cite{CRT,DonohoPol} rigorously proved (in a largedimensional and statistical context) that if the number of equations(measurements in the compressed sensing terminology) in the system isproportional to the length of the unknown vector then there is a sparsity(number of non-zero elements of the unknown vector) also proportional to thelength of the unknown vector such that $\ell_1$-optimization algorithm succeedsin solving the system. In more recent papers\cite{StojnicCSetamBlock09,StojnicICASSP09block,StojnicJSTSP09} we consideredunder-determined systems with the so-called \textbf{block}-sparse solutions. Ina large dimensional and statistical context in \cite{StojnicCSetamBlock09} wedetermined lower bounds on the values of allowable sparsity for any givennumber (proportional to the length of the unknown vector) of equations suchthat an $\ell_2/\ell_1$-optimization algorithm succeeds in solving the system.These lower bounds happened to be in a solid numerical agreement with what onecan observe through numerical experiments. Here we derive the correspondingupper bounds. Moreover, the upper bounds that we obtain in this paper match thelower bounds from \cite{StojnicCSetamBlock09} and ultimately make them optimal.

Optimality of $\ell_2/\ell_1$-optimization block-length dependent thresholds