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Automatic Relevance Determination in Nonnegative Matrix Factorization with the $(\beta)$-Divergence
July 2013 (vol. 35 no. 7)
pp. 1592-1605
V. Y. F. Tan, Inst. for Infocomm Res., A*STAR, Singapore, Singapore
C. Fevotte, Lab. Lagrange, Univ. de Nice Sophia Antipolis, Nice, France
This paper addresses the estimation of the latent dimensionality in nonnegative matrix factorization (NMF) with the β-divergence. The β-divergence is a family of cost functions that includes the squared euclidean distance, Kullback-Leibler (KL) and Itakura-Saito (IS) divergences as special cases. Learning the model order is important as it is necessary to strike the right balance between data fidelity and overfitting. We propose a Bayesian model based on automatic relevance determination (ARD) in which the columns of the dictionary matrix and the rows of the activation matrix are tied together through a common scale parameter in their prior. A family of majorization-minimization (MM) algorithms is proposed for maximum a posteriori (MAP) estimation. A subset of scale parameters is driven to a small lower bound in the course of inference, with the effect of pruning the corresponding spurious components. We demonstrate the efficacy and robustness of our algorithms by performing extensive experiments on synthetic data, the swimmer dataset, a music decomposition example, and a stock price prediction task.
Index Terms:
Bayesian methods,Linear programming,Cost function,Data models,Principal component analysis,Algorithm design and analysis,automatic relevance determination,Nonnegative matrix factorization,model order selection,majorization-minimization,group-sparsity
Citation:
V. Y. F. Tan, C. Fevotte, "Automatic Relevance Determination in Nonnegative Matrix Factorization with the $(\beta)$-Divergence," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 35, no. 7, pp. 1592-1605, July 2013, doi:10.1109/TPAMI.2012.240
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