  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, that 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.
