  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 $\ell_{2,\log}$-(pseudo) norm to enhance
the robustness of the proposed method.The $\ell_{2,\log}$-(pseudo) norm is
invariant, continuous, and differentiable.For the $\ell_{2,\log}$ 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.
