Almost optimality of orthogonal super greedy algorithms for incoherent dictionaries

Abstract

We investigate the efficiency of weak orthogonal super greedy algorithm (WOSGA) for [Formula: see text]-term approximation with respect to dictionaries which are [Formula: see text]-unconditional in arbitrary Hilbert space [Formula: see text] For an element [Formula: see text], let [Formula: see text] be the output of WOSGA after [Formula: see text] steps for some constant [Formula: see text]. We show that the residual [Formula: see text] can be bounded by a constant multiplying the error of best [Formula: see text]-term approximation to [Formula: see text] Moreover, we get an element [Formula: see text], through a simple postprocessing of [Formula: see text] by retaining its [Formula: see text] largest components in absolute value, which realizes near best [Formula: see text]-term approximation for [Formula: see text] Our results are obtained for dictionaries in [Formula: see text] which satisfies the weaker assumption than the RIP condition.