A fixed-point iterative schema for error minimization in k-sparse decomposition

Analogously to the well known greedy strategy called Orthogonal Matching Pursuit (OMP), we present a new algorithm to solve the sparse approximation problem over redundant dictionaries where the input signal is restricted to be a linear combination of k atoms or fewer from a fixed dictionary. The basic strategy of our method rests on a family of nonlinear mappings which results to be contractive in a interval close to zero. By iterating contractions and projections the method is able to extract the most significant components also for noisy signal which subsumes an ideal underlying signal having sufficiently sparse representation. For reasonable error level, the fixed point solution of such a iterative schema provides a sparse approximation containing only the nonzero terms characterizing the unique sparsest representation of the ideal noiseless sparse signal. The heuristic method so derived has been applied both to synthetic and real data. The former was generated by combining exact signals drawn by usual Bernoulli-Gaussian model and Gaussian noise; the later is taken by electrocardiogram (ECG) signals with application to the dictionary learning problem. In both cases the proposed method outperforms OMP method both regarding sparse approximation error and computation time.