Recovery and Convergence Rate of the Frank–Wolfe Algorithm for the m-Exact-Sparse Problem

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