Block-length dependent thresholds in block-sparse compressed sensing

Understand

One of the most basic problems in compressed sensing is solving an under-determined system of linear equations.

• Although this problem seems rather hard certain $\ell_1$-optimization algorithm appears to be very successful in solving it.
• The recent work of \cite{CRT,DonohoPol} rigorously proved (in a large dimensional and statistical context) that if the number of equations (measurements in the compressed sensing terminology) 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 $\ell_1$-optimization algorithm succeeds in solving the system.
• In more recent papers \cite{StojnicICASSP09block,StojnicJSTSP09} we considered the setup of the so-called \textbf{block}-sparse unknown vectors.