Elastic nonnegative matrix factorization

Abstract

Nonnegative matrix factorization (NMF) plays a vital role in data mining and machine learning fields. Standard NMF utilizes the Frobenius norm while robust NMF uses the robust ℓ2,1-norm to measure the quality of factorization, given the assumption of i.i.d Gaussian noise model and i.i.d Laplacian noise model, respectively. In this paper, we propose a novel elastic loss which is intercalated and adapted between Frobenius norm and ℓ2,1-norm. Inspired by this, we derive an elastic NMF model guided by the elastic loss with incorporating geometry manifold information while enforcing sparsity of coefficients at intra-cluster level via ℓ1,2-norm. The new formulation is more robust to noises while preserving the stronger capability of clustering. We propose an EM-like algorithm (using an auxiliary function) to solve the resultant optimization problem, whose convergence can be rigorously proved. The extensive experiments demonstrate the effectiveness of the novel elastic NMF model on benchmarks.