Bayesian Nonnegative Matrix Factorization with a Truncated Spike-and-Slab Prior

Non-negative matrix factorization (NMF) is a challenging problem due to its ill-posed nature. The key for the success of NMF is to exploit appropriate prior models for those two decomposed factor matrices. Although lots of effective sparsity-inducing prior models have been developed for NMF, they are often rooted in either ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</sub> regularization with p > 0, which only provide an approximation to the ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> sparsity, ultimately resulting in a sub-optimal solution. To address this problem, we propose a novel truncated spike-and-slab prior based Bayesian NMF method. Through integrating a Bernoulli distribution with a truncated Gaussian distribution together, the proposed prior is capable of imposing the exact ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> regularization as well as the non-negativity constraint on the factor matrices. Further, the proposed prior can be extended to robust NMF problem. Experimental results in blind source separation, face images representation and image denoising demonstrate the advantage of the proposed method.