A generalization of weighted sparse decomposition to negative weights.

Sparse solutions of underdetermined linear systems of equations are widely used in different fields of signal processing. This problem can also be seen as a sparse decomposition problem. Traditional sparse decomposition gives the same priority to all atoms for being included in the decomposition or not. However, in some applications, one may want to assign different priorities to different atoms for being included in the decomposition. This results to the so called "weighted sparse decomposition" problem [Babaie-Zadeh et al. 2012]. However, Babaie-Zadeh et al. studied this problem only for positive weights; but in some applications (e.g. classification) better performance can be obtained if some weights become negative. In this paper, we consider "weighted sparse decomposition" problem in its general form (positive and negative weights). A tight uniqueness condition and some applications for the general case will be presented.