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We discuss a general notion of "sparsity structure" and associated recoveriesof a sparse signal from its linear image of reduced dimension possiblycorrupted with noise. Our approach allows for unified treatment of (a) the"usual sparsity" and "usual $\ell_1$ recovery," (b) block-sparsity withpossibly overlapping blocks and associated block-$\ell_1$ recovery, and (c)low-rank-oriented recovery by nuclear norm minimization. The proposed recoveryroutines are natural extensions of the usual $\ell_1$ minimization used inCompressed Sensing. Specifically we present nullspace-type sufficientconditions for the recovery to be precise on sparse signals in the noiselesscase. Then we derive error bounds for imperfect (nearly sparse signal, presenceof observation noise, etc.) recovery under these conditions. In all of thesecases, we present efficiently verifiable sufficient conditions for the validityof the associated nullspace properties.

On unified view of nullspace-type conditions for recoveries associated with general sparsity structures