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In this paper we consider under-determined systems of linear equations thathave sparse solutions. This subject attracted enormous amount of interest inrecent years primarily due to influential works \cite{CRT,DonohoPol}. In astatistical context it was rigorously established for the first time in\cite{CRT,DonohoPol} that if the number of equations is smaller than but stilllinearly proportional to the number of unknowns then a sparse vector ofsparsity also linearly proportional to the number of unknowns can be recoveredthrough a polynomial $\ell_1$-optimization algorithm (of course, this assumingthat such a sparse solution vector exists). Moreover, the geometric approach of\cite{DonohoPol} produced the exact values for the proportionalities inquestion. In our recent work \cite{StojnicCSetam09} we introduced analternative statistical approach that produced attainable values of theproportionalities. Those happened to be in an excellent numerical agreementwith the ones of \cite{DonohoPol}. In this paper we give a rigorous analyticalconfirmation that the results of \cite{StojnicCSetam09} indeed match those from\cite{DonohoPol}.

A rigorous geometry-probability equivalence in characterization of $\ell_1$-optimization