Infinite Positive Semidefinite Tensor Factorization by K. Yoshii, R. Tomioka, D. Mochihashi, and M. Goto Infinite Positive Semidefinite Tensor Factorization for Source Separation of Mixture Signals Kazuyoshi Yoshii Ryota Tomioka Daichi Mochihashi Masataka Goto 1) 1) 3) 2) 1) National Institute of Advanced Industrial Science and Technology (AIST) 2) The University of Tokyo 3)The Institute of Statistical Mathematics (ISM)
Infinite Positive Semidefinite Tensor Factorization by K. Yoshii, R. Tomioka, D. Mochihashi, and M. Goto A special case: log-determinant PSDTF (LD-PSDTF) We proposed positive semidefinite tensor factorization (PSDTF) is a naive extension of NMF Take-Home Messages that can deal with an infinite number of bases • – Tensor extension of nonnegative matrix factorization (NMF) • Nonnegative tensor factorization (NTF) – Bayesian nonparametrics • The gamma process is used for Bayesian PSDTF • – Elegant variational inference • Closed-form MU and VB updates were derived – Various applications • Single-channel audio source separation • Multi-channel EEG signal analysis
Infinite Positive Semidefinite Tensor Factorization by K. Yoshii, R. Tomioka, D. Mochihashi, and M. Goto Nonnegative Matrix Factorization Each nonnegative vector is approximated Observed matrix Basis matrix Activation matrix Vector-wise factorization Bregman divergence: Kullback-Leibler divergence: Itakura-Saito divergence: → Minimize • by a convex combination of nonnegative vectors (bases)
Infinite Positive Semidefinite Tensor Factorization by K. Yoshii, R. Tomioka, D. Mochihashi, and M. Goto A Major Limitation of NMF The elements of each basis vector are assumed to be independent Element-wise representation Problem: STFT cannot completely decorrelate frequency bins because finite windows are used for analyzing non-stationary signals Gamma priors are placed in an element-wise manner The cost function is defined in an element-wise manner • – The correlations between those elements are ignored
Infinite Positive Semidefinite Tensor Factorization by K. Yoshii, R. Tomioka, D. Mochihashi, and M. Goto … Positive Semidefinite Tensor Factorization Each positive semidefinite matrix is approximated … … … … Observed tensor Basis tensor Activation matrix Matrix-wise factorization Bregman matrix divergence can be used as a cost function • by a convex combination of positive semidefinite matrices (bases)
Infinite Positive Semidefinite Tensor Factorization by K. Yoshii, R. Tomioka, D. Mochihashi, and M. Goto PSDTF: A Natural Extension of NMF Bayesian infinite Nonparametric Positive Semidefinite Tensor Factorization (PSDTF) extension feasible Nonnegative Matrix Factorization (NMF) • – Vector-wise factorization – Bregman divergence • Kullback-Leibler (KL) divergence • Itakura-Saito (IS) divergence • – Matrix-wise factorization – Bregman matrix divergence • von Neumann (vN) divergence • Log-determinant (LD) divergence
Infinite Positive Semidefinite Tensor Factorization by K. Yoshii, R. Tomioka, D. Mochihashi, and M. Goto 35 LD-PSDTF IS-NMF K-NMF SAR SIR SDR 30 Single-Channel Audio Source Separation 25 20 15 10 LD-PSDTF outperformed KL-NMF and IS-NMF [dB] • – Tested on a toy mixture signal consisting of three piano sounds
Infinite Positive Semidefinite Tensor Factorization by K. Yoshii, R. Tomioka, D. Mochihashi, and M. Goto A special case: log-determinant PSDTF (LD-PSDTF) We proposed positive semidefinite tensor factorization (PSDTF) is a naive extension of NMF Take-Home Messages that can deal with an infinite number of bases Another special case: von-Neumann PSDTF (vN-PSDTF) • – Tensor extension of nonnegative matrix factorization (NMF) • Nonnegative tensor factorization (NTF) – Bayesian nonparametrics • The gamma process is used for Bayesian PSDTF • – Elegant variational inference • Closed-form MU and VB updates were derived • – This is worth investigating (closed-form solution exists?)
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