## Block-length dependent thresholds in block-sparse compressed sensing

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Stojnic, Mihailo

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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. In a large dimensional and
statistical context, we determined sharp lower bounds on the values of
allowable sparsity for any given number (proportional to the length of the
unknown vector) of equations such that an $\ell_2/\ell_1$-optimization
algorithm succeeds in solving the system. The results established in
\cite{StojnicICASSP09block,StojnicJSTSP09} assumed a fairly large block-length
of the block-sparse vectors. In this paper we consider the block-length to be a
parameter of the system. Consequently, we then establish sharp lower bounds on
the values of the allowable block-sparsity as functions of the block-length.

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Computer Science - Information Theory