Abstract
It has been known for a while that lscr <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> -norm relaxation can in certain cases solve an under-determined system of linear equations. Recently, 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 lscr <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> -norm relaxation succeeds in solving the system. In this paper we consider a modification of this standard setup, namely the case of so-called approximately block-sparse unknown vectors. We determine sharp lower bounds on the values of allowable approximate block-sparsity for any given number (proportional to the length of the unknown vector) of equations. Obtained lower bounds on the allowable sparsity are as expected functions of a parameter used to describe how close the approximately block-sparse unknown vectors are to the ideally block-sparse ones.
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