  This paper studies the effect of discretizing the parametrization of a
dictionary used for Matching Pursuit decompositions of signals. Our approach
relies on viewing the continuously parametrized dictionary as an embedded
manifold in the signal space on which the tools of differential (Riemannian)
geometry can be applied. The main contribution of this paper is twofold. First,
we prove that if a discrete dictionary reaches a minimal density criterion,
then the corresponding discrete MP (dMP) is equivalent in terms of convergence
to a weakened hypothetical continuous MP. Interestingly, the corresponding
weakness factor depends on a density measure of the discrete dictionary.
Second, we show that the insertion of a simple geometric gradient ascent
optimization on the atom dMP selection maintains the previous comparison but
with a weakness factor at least two times closer to unity than without
optimization. Finally, we present numerical experiments confirming our
theoretical predictions for decomposition of signals and images on regular
discretizations of dictionary parametrizations.
