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This article provides a new type of analysis of a compressed-sensing basedtechnique for recovering column-sparse matrices, namely minimization of the$\ell_{1,2}$-norm. Rather than providing conditions on the measurement matrixwhich guarantees the solution of the program to be exactly equal to the groundtruth signal (which already has been thoroughly investigated), it presents acondition which guarantees that the solution is approximately equal to theground truth. Soft recovery statements of this kind are to the best knowledgeof the author a novelty in Compressed Sensing. Apart from the theoreticalanalysis, we present two heuristic proposes how this property of the$\ell_{1,2}$-program can be utilized to design algorithms for recovery ofmatrices which are sparse and have low rank at the same time.

Soft Recovery Through $\ell_{1,2}$ Minimization with Applications in Recovery of Simultaneously Sparse and Low-Rank Matrice