This paper studies the effect of discretizing the parametrization of a dictionary used for matching pursuit (MP) 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.
In @cite_33 , the author uses a 4-dimensional Gaussian chirp dictionary to analyze 1-D signals with MP algorithm. He develops a fast procedure to find the best atom of this dictionary in the representation of the current MP residual by applying a two-step search. First, by setting the chirp rate parameter to zero, the best common Gabor atom is found with full search procedure taking advantage of the FFT algorithm. Next, a ridge theorem proves that starting from this Gabor atom, the best Gaussian chirp atom can be approximated with a controlled error. The whole method is similar to the development of our optimized matching pursuit since we start also from a discrete parametrization to find a better atom in the continuous one. However, our approach is more general since we are not restricted to a specific dictionary. We use the intrinsic geometry of any smooth dictionary manifold to perform a optimization driven by a geometric gradient ascent.
@cite_33
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We introduce a modified matching pursuit algorithm, called fast ridge pursuit, to approximate N-dimensional signals with M Gaussian chirps at a computational cost O(MN) instead of the expected O(MN sup 2 logN). At each iteration of the pursuit, the best Gabor atom is first selected, and then, its scale and chirp rate are locally optimized so as to get a "good" chirp atom, i.e., one for which the correlation with the residual is locally maximized. A ridge theorem of the Gaussian chirp dictionary is proved, from which an estimate of the locally optimal scale and chirp is built. The procedure is restricted to a sub-dictionary of local maxima of the Gaussian Gabor dictionary to accelerate the pursuit further. The efficiency and speed of the method is demonstrated on a sound signal.