( ) (Harmonic-Temporal Clustering; HTC) [1], [2] ( ) ( ) [4] HTC. (Non-negative Matrix Factorization; NMF) [3] [5], [6] [7], [8]

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1 , Product of ExpertsPoE) 1. ) Harmonic-Temporal Clustering; HTC) [1], [] ) ) HTC Non-negative Matrix Factorization; NMF) [3] 1 Graduate School of Information Science and Technology, The University of Tokyo NTT NTT Comunication Science Labolatories, Nippon Telegraph and Telephone Coporation [4] [5], [6] [7], [8] Dirichlet Poisson N, Dir, Pois c 014 Information Processing Society of Japan 1

2 ..1 [1] ) k n nθ k u) + φ a u) f k u) = N a u)e jnθ ku)+φ ) n=1 u φ f k u) W k x, t) ψ α,t u) ψ α,t u) = 1 ) u t ψ ) πα α 1) x t α ψu) 1 f k u) W k log 1 ) α, t = f k u), ψ α,t u) 3) F k ω) f k u) Fourier Ψ α,t ω) ψ α,t u) Fourier Parseval f k u), ψ α,t u) = F k ω), Ψ α,t ω) 4) ψu) Fourier Ψω) ) Fourier Ψ α,t ω) = Ψαω)e jωt 5) 4) W k log 1 ) α, t = F k ω)ψ αω)e jωt dω 6) a u) ã θ k u) θ k u) θ k u) = µ k u µ k µ k u) f k u) Fourier F k ω) = 1 N π = π n=1 ã e jn µ ku+φ ) e jωu du 7) N ã e jφ δω n µ k ) n=1 f k u) W k x, t) x = log 1 α W k x, t) = n a t)ψ ne x µ k t))e jnθ kt)+φ ) 8) Ψ ) ln ω) exp 4σ ω > 0) Ψω) = 9) 0 ω 0) Ω k t) = ln µ k t) W k x, t) W k x, t) = n a t)e x Ω k t) ln n) 4σ e jnθ kt)+φ ) 10) σ n, n n n ) ) W k x, t) W k x, t) n a t) e x Ω k t) ln n) σ 11) HTC Gaussian mixture model; GMM). NMF NMF a t) t n a t) = w U k t)/ πσ 1) W k x, t) W k x, t) = H k x, t)u k t) 13) H k x, t) := n w t) ln n) e x Ω k σ 14) πσ H k x, t) GMM k t U k t) w U w = 1 15) n w.4 HTC NMF c 014 Information Processing Society of Japan

3 H k NMF [10], [11] 14) Ω k t) [1], [13] U k t) GMM HTC[1], [] NMF HTC x l l = 1,..., L) t i i = 1,..., I) Y x l, t i ) HTC NMF 13) Xx l, t i ) = k H k x l, t i )U k t i ) 16) Xx l, t i ) 1) Y x l, t i ) Y x l, t i ) PoisY x l, t i ); Xx l, t i )) l, i) 17) Xx l, t i ) I Y X.4 Xx l, t i ) H k x l, t i ) 14) k, l, i H k x l, t i ) 16) NMF [9] H k x l, t i ) NMF [3] 3.1 [4] Products of Experts[14] ) 3.1 k Ω k t i ) q g q s q g Ω k t i )) = N Ω k t i ); m k, ν k) 18) q c Ω k t i ) Ω k t i 1 )) = N Ω k t i ); Ω k t i 1 ), τ k ) 19) m k, ν k k τ k q c c 014 Information Processing Society of Japan 3

4 q g m k q g, q c Ω k t) pω k t i ) Ω k t i 1 )) q g Ω k t i )) αg q c Ω k t i ) Ω k t i 1 )) αc 0) α g, α c q g, q c 3.. U k t) C B k t i ) i B kt i ) = 1) A k k A k = 1) U k t i ) = CA k B k t i ), 1) A := [A 1,..., A K ] DirA; β) ) B k := [B k t 1 ),..., B k t I )] DirB k ; γ k ) 3) A β B k γ k Y := {Y x l, t i )} l,i MAP Θ = {w, Ω k t i ), A k, B k t i ), C} i argmax Θ ln pθ Y ) = argmax ln py Θ) + ln pθ)) Θ 4) ln pθ Y ) Θ ln py Θ) pθ) ln py Θ) ) = c Y x l, t i ) ln H k x l, t i )U k t) k k H k x l, t i )U k t) ) 5) ln pθ) = c α g ln q g Ω k t i )) + α c ln q s Ω k t i ) Ω k t i 1 ))) k,i + i ln DirA; β) + k ln DirB k ; γ k ) 6) = c 5) H k x l, t i )U k t i ),k = l,i, U k t i ) w exp x ) l Ω k t) ln n) πσ σ 7) 1 exp x Ω ) kt i ) ln n) πσ σ dx = 1 8) U k t i ) w exp x ) l Ω k t i ) ln n) πσ σ l,n U kt i ) X 0 9) X 0 4) arg max Θ JΘ) JΘ) Θ 5) k, n [15] JΘ) Θ JΘ) 5) Jensen ) Y x l, t i ) ln w ϕ x l, t i )U k t) Y x l, t i ) λ, U k t i ) ln w ϕ x l, t i ) λ, 30) λ, [0, 1] i, l, λ, = 1 λ,k = U k t i )w ϕ x l, t i ) n,k U kt i )w ϕ x l, t i ), 31) JΘ) J + Θ, {λ, } ) = c Y x l, t i ) λ, U k t i ) ln w ϕ x l, t i ) λ, U kt i ) X 0 + k,i + i α g ln q g Ω k t i )) + α c ln q s Ω k t i ) Ω k t i 1 ))) ln DirA; β) + k ln DirB k ; γ k ) 3) c 014 Information Processing Society of Japan 4

5 I- NMF 3) 4. 31) J + Θ, {λ,k, ζ, } ) 0 C Y C = X 0 Y x l, t i ) 33) l,i w Y x l, t i )λ, l,i,n Y x 34) l, t i )λ, αc Ω k τ D D + α ) 1 g ν E I + diag p l, ) m k α ) c ν 1 I + x l ln n) p l, 35) l,n l,i,n A k Y x l, t i )λ, + β k 1 k l,i,n Y x 36) l, t i )λ, + β k 1) l,n B k t i ) Y x l, t i )λ, + γ i,k 1 t l,n Y x 37) l, t i )λ, + γ i,k 1) diag 1 I I 1 E I I Ω k p l, Ω k := [Ω k t 1 ),..., Ω k t I )] 38) p l,,i = Y x l, t i )λ, σ 39) p l, := [p l,,1,..., p l,,i ] I I D D = ) MATLAB 3 D,F,A ) Ω k t i ) 5.1 RWC [16] D F A 16 khz I- NMF 3, 100 ) [17] t i t i 1 = 16 ms [1] x 1 = ln55), x l x l 1 = ln)/10 N, K, τ, ν, σ, α g, α c ) = 8, 73, 10 4, 1.5, 10 4, 0.0, 1, 1) γ k = )1 I β = )1 K 10. 1). NMF D D β = β = D ) 6. NMF HTC c 014 Information Processing Society of Japan 5

6 4 ), ) Product of Experts JSPS [1] Hirokazu Kameoka, Statistical approach to multipitch analysis, Ph.D. thesis, University of Tokyo, 007. [] Hirokazu Kameoka, Takuya Nishimoto, Shigeki Sagayama, A Multipitch Analyzer Based on Harmonic Temporal Structured Clustering, IEEE Transactions on Audio, Speech and Language Processing, Vol. 15, No. 3, pp , Mar [3] Paris Smaragdis, and Judith C. Brown, Non-Negative Matrix Factorization for Polyphonic Music Transcription, In Proc. the IEEE Workshop on Applications of Signal Processing to Audio and Acoustics, New York, USA, 003. [4] [5] Paris. Smaragdis, and Gautham J. Mysore, Separation by humming : Userguided sound extraction from monophonic mixtures, In Proc. the IEEE Workshop on Applications of Signal Processing to Audio and Acoustics, pp. 69 7, 009. [6] Alexey Ozerov, Cédric Févotte, Raphaël Blouet, and Jean-Louis Durrieu, Multichannel nonnegative tensor factorization with structured constraints for user-guided audio source separation, In Proc. the IEEE International Conference on Acoustics, Speech, and Signal Processing, pp , 011. [7] Romain Hennequin, Bertrand David, and Roland Badeau, Score informed audio source separation using a parametric model of non-negative spectrogram, In Proc. the IEEE International Conference on Acoustics, Speech, and Signal Processing, pp , 011. [8] Umut Simsekli, A. Taylan Cemgil, Score guided musical source separation using generalized coupled tensor factorization. In Proc. European Signal Processing Conference, pp , 01. [9] Masahiro Nakano, Jonathan Le Roux, Hirokazu Kameoka, Nobutaka Ono, Shigeki Sagayama, Infinite- State Spectrum Model for Music Signal Analysis, In Proc. the IEEE International Conference on Acousitcs, Speech, and Signal Processing, 011. [10] Stanislaw A. Raczyński, Nobutaka Ono, and Shigeki Sagayama, Multipitch analysis with harmonic nonnegative matrix approximation, In Proc. The International society for Music Information Retrieval, pp , 007. [11] Emmanuel Vincent, Nancy Bertin, and Roland Badeau, Harmonic and inharmonic nonnegative matrix factorization for polyphonic pitch transcription, In Proc. The IEEE International Conference on Acoustics, Speech, and Signal Processing, pp , 008. [1] Kazuyoshi Yoshii, and Masataka Goto, Infinite Latent Harmonic Allocation: A Nonparametric Bayesian Approach to Multipitch Analysis, In Proc. The International society for Music Information Retrieval, pp , 010. [13], 75, 4T-8, 013. [14] Geoffrey E. Hinton, Training products of experts by minimizing contrastive divergence. Neural computation, 148), pp , 00. [15],,,,, 006-MUS-66-13, pp , Aug [16] Masataka Goto, Development of the RWC Music Database, In Proc. of the 18th International Congress on Acoustics ICA 004), pp. I , April 004. [17],,,,, , 11, 008. c 014 Information Processing Society of Japan 6