Non-negative sparse decomposition based on constrained smoothed ℓ0 norm

• To obtain sparse solution with coefficients in a specified range, we propose a modified version of L0 norm.
• The sufficient condition for all coefficients to be inside the desired range is provided.
• We propose a coarse to fine approach to solve the Constrained Smoothed L0 optimization problem.
• The proposed algorithm is evaluated on both simulated and real data.

Sparse decomposition of a signal over an overcomplete dictionary has many applications including classification. One of the sparse solvers that has been proposed for finding the sparse solution of a spare decomposition problem (i.e., solving an underdetermined system of equations) is based on the Smoothed L0 norm (SL0). In some applications such as classification of visual data using sparse representation, the coefficients of the sparse solution should be in a specified range (e.g., non-negative solution). This paper presents a new approach based on the Constrained Smoothed L0 norm (CSL0) for solving sparse decomposition problems with non-negative constraint. The performance of the new sparse approach is evaluated on both simulated and real data. For the simulated data, the mean square error of the solution using the CSL0 is comparable to state-of-the-art sparse solvers. For real data, facial expression recognition via sparse representation is studied where the recognition rate using the CSL0 is better than other solver methods (in particular is about 4% better than the SL0).