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We study the properties of the Frank-Wolfe algorithm to solve them-EXACT-SPARSE reconstruction problem, where a signal y must be expressed as asparse linear combination of a predefined set of atoms, called dictionary. Weprove that when the signal is sparse enough with respect to the coherence ofthe dictionary, then the iterative process implemented by the Frank-Wolfealgorithm only recruits atoms from the support of the signal, that is thesmallest set of atoms from the dictionary that allows for a perfectreconstruction of y. We also prove that under this same condition, there existsan iteration beyond which the algorithm converges exponentially.

Recovery and convergence rate of the Frank-Wolfe Algorithm for the m-EXACT-SPARSE Problem