A sparse optimization approach to supervised NMF based on convex analytic method

In this paper, we propose a novel scheme to supervised nonnegative matrix factorization (NMF). We formulate the supervised NMF as a sparse optimization problem assuming the availability of a set of basis vectors, some of which are irrelevant to a given matrix to be decomposed. The proposed scheme is presented in the context of music transcription and musical instrument recognition. In addition to the nonnegativity constraint, we introduce three regularization terms: (i) a block ℓ1 norm to select relevant basis vectors, and (ii) a temporal-continuity term plus the popular ℓ1 norm to estimate correct activation vectors. We present a state-of-the-art convex-analytic iterative solver which ensures global convergence. The number of basis vectors to be actively used is obtained as a consequence of optimization. Simulation results show the efficacy of the proposed scheme both in the case of perfect/imperfect basis matrices.

[1]  H. Sebastian Seung,et al.  Learning the parts of objects by non-negative matrix factorization , 1999, Nature.

[2]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[3]  Roland Badeau,et al.  Multipitch Estimation of Piano Sounds Using a New Probabilistic Spectral Smoothness Principle , 2010, IEEE Transactions on Audio, Speech, and Language Processing.

[4]  Michael Elad,et al.  Sparse and Redundant Representations - From Theory to Applications in Signal and Image Processing , 2010 .

[5]  Mert Bay,et al.  Evaluation of Multiple-F0 Estimation and Tracking Systems , 2009, ISMIR.

[6]  Perry R. Cook,et al.  Bayesian Nonparametric Matrix Factorization for Recorded Music , 2010, ICML.

[7]  Andrzej Cichocki,et al.  Nonnegative Matrix and Tensor Factorization T , 2007 .

[8]  Heinz H. Bauschke,et al.  Convex Analysis and Monotone Operator Theory in Hilbert Spaces , 2011, CMS Books in Mathematics.

[9]  H. Sebastian Seung,et al.  Algorithms for Non-negative Matrix Factorization , 2000, NIPS.

[10]  Masataka Goto,et al.  Development of the RWC Music Database , 2004 .

[11]  Jordi Vitrià,et al.  Analyzing non-negative matrix factorization for image classification , 2002, Object recognition supported by user interaction for service robots.

[12]  Tuomas Virtanen,et al.  Monaural Sound Source Separation by Nonnegative Matrix Factorization With Temporal Continuity and Sparseness Criteria , 2007, IEEE Transactions on Audio, Speech, and Language Processing.

[13]  Daniel P. W. Ellis,et al.  Transcribing Multi-Instrument Polyphonic Music With Hierarchical Eigeninstruments , 2011, IEEE Journal of Selected Topics in Signal Processing.

[14]  P. Smaragdis,et al.  Non-negative matrix factorization for polyphonic music transcription , 2003, 2003 IEEE Workshop on Applications of Signal Processing to Audio and Acoustics (IEEE Cat. No.03TH8684).

[15]  P. Paatero,et al.  Positive matrix factorization: A non-negative factor model with optimal utilization of error estimates of data values† , 1994 .

[16]  Guillaume Lemaitre,et al.  Real-Time Detection of Overlapping Sound Events with Non-Negative Matrix Factorization , 2013 .

[17]  C. Févotte,et al.  Automatic Relevance Determination in Nonnegative Matrix Factorization , 2009 .

[18]  Isao Yamada,et al.  Minimizing the Moreau Envelope of Nonsmooth Convex Functions over the Fixed Point Set of Certain Quasi-Nonexpansive Mappings , 2011, Fixed-Point Algorithms for Inverse Problems in Science and Engineering.