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In this paper we consider random linear under-determined systems withblock-sparse solutions. A standard subvariant of such systems, namely,precisely the same type of systems without additional block structuringrequirement, gained a lot of popularity over the last decade. This is of coursein first place due to the success in mathematical characterization of an$\ell_1$ optimization technique typically used for solving such systems,initially achieved in \cite{CRT,DOnoho06CS} and later on perfected in\cite{DonohoPol,DonohoUnsigned,StojnicCSetam09,StojnicUpper10}. The successthat we achieved in \cite{StojnicCSetam09,StojnicUpper10} characterizing thestandard sparse solutions systems, we were then able to replicate in a sequenceof papers\cite{StojnicCSetamBlock09,StojnicUpperBlock10,StojnicICASSP09block,StojnicJSTSP09}where instead of the standard $\ell_1$ optimization we utilized its an$\ell_2/\ell_1$ variant as a better fit for systems with block-sparsesolutions. All of these results finally settled the so-called threshold/phasetransitions phenomena (which naturally assume the asymptotic/large dimensionalscenario). Here, in addition to a few novel asymptotic considerations, we alsotry to raise the level a bit, step a bit away from the asymptotics, andconsider the finite dimensions scenarios as well.

Random linear under-determined systems with block-sparse solutions -- asymptotics, large deviations, and finite dimensions