Dictionary-Sparse Recovery via Thresholding-Based Algorithms

It is shown that the iterative hard thresholding and hard thresholding pursuit algorithms provide the same theoretical guarantees as $$\ell _1$$ℓ1-minimization for the recovery from imperfect compressive measurements of signals that have almost sparse analysis expansions in a fixed dictionary. Unlike other signal space algorithms targeting the recovery of signals with sparse synthesis expansions, the ability to compute (near) best approximations by synthesis-sparse signals is not necessary. The results are first established for tight frame dictionaries, before being extended to arbitrary dictionaries modulo an adjustment of the measurement process.