Log-based sparse nonnegative matrix factorization for data representation

Nonnegative matrix factorization (NMF) has been widely studied in recent years due to its effectiveness in representing nonnegative data with parts-based representations. For NMF, a sparser solution implies better parts-based representation. However, current NMF methods do not always generate sparse solutions. In this paper, we propose a new NMF method with log-norm imposed on the factor matrices to enhance the sparseness. Moreover, we propose a novel column-wisely sparse norm, named ${\ell }_{2,\text{log}}$ -(pseudo) norm to enhance the robustness of the proposed method. The ${\ell }_{2,\text{log}}$ -(pseudo) norm is invariant, continuous, and differentiable. For the ${\ell }_{2,\text{log}}$ regularized shrinkage problem, we derive a closed-form solution, which can be used for other general problems. Efficient multiplicative updating rules are developed for the optimization, which theoretically guarantees the convergence of the objective value sequence. Extensive experimental results confirm the effectiveness of the proposed method, as well as the enhanced sparseness and robustness.