Matching pursuit shrinkage in Hilbert spaces

This paper contains the research on a hybrid algorithm combining the Matching Pursuit (MP) and the wavelet shrinkage. In this algorithm, we propose to shrink the scalar product of the element which best correlates with the residue before modifying. The study concerns a broad family of shrinkage functions. Using weak properties of these shrinkage functions, we show that the algorithm converges towards the orthogonal projection of the data on the linear space generated by the dictionary, modulo a precision characterized by the shrinkage function. In the deterministic settings, under a mild assumption on the shrinkage function (for instance, the hard shrinkage satisfies this assumption), this algorithm converges in a finite time which can be estimated from the properties of the shrinkage function. Experimental results show that in the presence of noise, the new algorithm does not only outperform the regular MP, but also behaves better than some other classical Greedy methods and Basis Pursuit Denoising model when used for detection.