Non-negative Matrix Factorization, NMF [5] NMF. [1 3] Bregman [4] Harmonic-Temporal Clustering, HTC [2,3] 1,2,b) NTT

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1 1,a) 1,2,b) 1. [1 3] Bregman [4] Harmonic-Temporal Clustering, HTC [2,3] NTT a) Tomohio Naamura@ipc.i.u-toyo.ac.jp b) ameoa@hil.t.u-toyo.ac.jp/ameoa.hiroazu@lab.ntt.co.jp Non-negative Matrix Factorization, NMF [5] NMF c 1959 Information Processing Society of Japan 1

2 Harmonic-Temporal Factor Decomposition, HTFD [6 8]. [9] HTFD [10,11] [12,13] R, C, j := [3] n = 1, 2,..., N 1 n nθ (u) R a,n (u) C f (u) = N a,n (u)e j(nθ (u)+φ,n ) u R φ,n R f (u) ψ α,t (u) 1 ( u t ) ψ α,t (u) = ψ 2πα α α > 0 t R ψ(u) 1 f (u) W (ln 1 α, t) = (1) (2) N a,n (u)e j(nθ (u)+φ,n ) ψ α,t(u)du (3) ψ α,t(u) t W (ln 1 α, t) t θ (u) a,n (u) θ (u) θ (t)+ θ (t)(u t), a,n (u) a,n (t) Parseval x := ln(1/α) F 0 Ω (t) = ln θ (t)w (x, t) N a,n (t)ψ (ne x+ω(t) )e j(nθ (t)+φ,n ), (4) θ (u) F 0 ψ Fourier Ψ ω = 1 [3] (ln ω)2 e 4σ 2 (ω > 0) Ψ(ω) =. (5) 0 (ω 0) σ Ψ(ω) ln ω (5) W (x, t) W (x, t) = N a,n (t)e (x Ω (t) ln n)2 4σ 2 e j(nθ (t)+φ,n ). (6) W (x, t) 2 W (x, t) 2 N a,n (t) 2 e (x Ω (t) ln n)2 2σ 2 (7) HTC [3] t t x t m (m = 0, 1,..., M 1) x l (l = 0, 1,..., L 1) Y l,m := Y(x l, t m ) Ω,m := Ω (t m ) a,n,m := a,n (t m ) 2.2 (7) [14] (1) i t m (7) c 1959 Information Processing Society of Japan 2

3 f,m [i] f,m [i] P β,m [0] f,m [i] = P β,m [p] f,m [i p] + ϵ,m [i], (8) p=1 β,m [p] (p = 0, 1,..., P) β,m [p] ϵ,m [i] 2.1 f,m [i] F 0 e Ω,m ϵ,m [i] F 0 e Ω,m ϵ,m [i] ϵ,m [i] = N v,n,m e jneω,m iu 0, (9) u 0 > 0 v,n,m C n f,m [i] Fourier discrete-time Fourier transform, DTFT DTFT f,m [i] = B,m (z) := N v,n,m e jne Ω,m iu0 (10) B,m (e jneω,m u 0) P β,m [p]z p (11) p=0 f,m [i] (10) (1) 2.1 a,n,m = v,n,m B,m (e jneω,m u 0) 2.3 (12) NMF F 0 a,n,m v,n,m a,n,m = w,n,m U,m, v,n,m = w,n,m U,m. (13) w,n,m w,n,m U,m t m U,m,m U,m = 1 1/ B,m (e jω ) 2 β,m [p] B,m (z) m (12) w,n,m = w,n,m B (e jneω,m u 0) 2.4 (14) C,l,m C,l,m =H,l,m U,m, (15) N H,l,m = w 2 (x l Ω,m ln n)2,n,m e 2σ 2 (16) HTC NMF X l,m X l,m = K C,l,m (17) =1 1X l,m Y l,m X l,m Poisson Y l,m Pois(Y l,m ; X l,m ) = XY l,m l,m e X l,m Γ(Y l,m ) (18) X l,m I X l,m Y l,m [14] w,n,m 0 ν 2 (14) w,n,m Rayleigh ( ) ν w,n,m Rayleigh w,n,m ; B (e jneω,m u 0) w,n,m = (ν/ B (e jneω,m u 0) ) 2 e w2,n,m /(2(ν/ B (e jneω,m u0 ) ) 2). 2.5 (19) X l,m (16) H,l,m c 1959 Information Processing Society of Japan 3

4 F 0 2 Ω Ω q l (Ω ), q g (Ω ) q g (Ω ) = N(Ω ; µ 1 M, ξ 2 I M), (20) 1 HTFD X l,m NMF [15] H,l,m NMF H,l,m NMF [16, 17] (16) Ω,m [18] U,m HTC [2, 3] Products of Experts (PoE) [19] 3.2 Ω,m U,m F 0 F 0 q l (Ω ) = N(Ω ; 0 M, τ 2 D 1 ), (21) D = (22) N(Ω ; µ, Σ) µ Σ M 1 M 1 M 0 M M I M M M 2 Ω p(ω ) q l (Ω ) α l q g (Ω ) α gl (23) α l, α g q g (Ω ), q g (Ω ) U,m U,m NMF U,m = R A,m R := m U,m ( R = 1) A,m := U,m /R ( m A,m = 1) R := [R 0, R 1,..., R K 1 ], A := [A,0, A,1,..., A,M 1 ] R Dir(R; γ (R) ), A Dir(A ; γ (A) ) (24) γ (R) := [γ (R) 1,..., γ(r) K ] R γ (R) R γ (A) := [γ (A),1,..., γ(a),m ] A A 4. Y p(θ Y) p(y Θ)p(Θ) w, Θ := {Ω, R, A} c 1959 Information Processing Society of Japan 4

5 J(Θ) := ln p(y Θ) + ln p(θ) (25) w := {w,n,m },n,m W ln p(y Θ) = ln Pois(Y l,m ; X l,m ) W l,m ( ) ν Rayleigh w,n,m ; dw B (e jneω,m u 0),n,m (26) ln p(θ) = ln p(ω ) + ln p(r) + ln p(a ). (27) (26) w J(Θ) Θ J(Θ) (26) Jensen ln p(y Θ) q(w) Y l,m ln X l,m X l,m + W Y l,m l,m l,m l,m + ln Rayleigh(w,n,m ; ν/b (e jneω,m u 0 ) ),n,m Y l,m ln q(w)) dw (28) q(w) q(w)dw = 1, q(w) 0. W q(w) w E q(w)[w 2 ] := W q(w)w2 dw (17) X l,m, n (28) 1 (28) 1 Jensen Y l,m ln X l,m Y l,m,n (x l Ω,m ln n) 2 λ,n,l,m ln w2,n,m e 2σ 2 U,m, λ,n,l,m (29) λ := {λ,n,l,m },n,l,m λ,n,l,m 0,,n λ,n,l,m = 1 (x l Ω,m ln n) 2 λ,n,l,m = w2,n,m e 2σ 2 U,m (30) X l,m J(Θ) J + (λ, q(w), Θ) J + (λ, q(w), Θ) (x l Ω,m ln n) 2 = E q(w) Y l,m λ,n,l,m ln w2,n,m e 2σ 2 U,m c λ l,m,n,n,l,m X l,m + ln l,m p(w β, Ω) q(w) + ln p(θ). (31) = c (31) 2 x x < x 0 x L 1 < x X l,m 1 X(x, t m )dx x l = 2πσ R A,m w 2,n,m. (32) x (31) 2 Ω,m J + (λ, q(w), Θ) J ++ (λ, q(w), Θ) Θ J ++ (λ, q(w), Θ) 0 [7] F 0 HTFD F 0 RWC [20] D 4 F4 A 4 16 Hz w,m,n HTFD [6] 14.6 ms 55 Hz 7040 Hz 10 cent [21] (5) σ = 0.02 N = 8 K = 73 µ A1 55 Hz A 7 γ (A) = ( )1 I τ = 0.83 v = 1.25 α g = α s = 1 γ (R) = ( )1 K 2 F 0 NMF 3 I F 0 2 (a) HTFD n c 1959 Information Processing Society of Japan 5

6 (a) HTFD F 0 (b) NMF 2 HTFD [6] NMF [8] D 4 F4 A 4 A Pitch D Pitch A D (a) 4 F 0 HTFD Harmonic NMF HTFD HTFD [6] HTFD HTFD+ [7] SNR (b) 3 (a) (b) HTFD A3 A 4 [8] D4 2 (b) F γ (R) γ (R) (a) D4 D4 γ (R) c 1959 Information Processing Society of Japan 6

7 5.3 HTFD [6] F 0 HTFD NMF [16, 17] Harmonic NMF C,l,m / C,l,m w q(w) w 2 E q(w)[w 2 ] RWC [20] RM-C001 RM-C MIDI FluidSynth [22] 16 Hz 14.6 ms N = 20 τ = 1.0 γ (A) = ( )1 I γ (R) = K P = 20, ν = Harmonic NMF 100 HTFD HTFD+ 20 signal-to-noise ratio SNR 4 Harmonic NMF HTFD SNR 0.02 db MIDI F 0 HTFD+ Harmonic NMF HTFD SNR 0.80 db 0.78 db 6. NMF HTC HTFD PoE JSPS [1] Hu, G. and Wang, D. L.: An auditory scene analysis approach to monaural speech segregation, Topics in Acoust. Echo and Noise Contr., pp (2006). [2] Kameoa, H., Nishimoto, T. and Sagayama, S.: A Multipitch Analyzer Based on Harmonic Temporal Structured Clustering, IEEE Trans. Acoust., Speech, and Language Process., Vol. 15, No. 3, pp (2007). [3] Kameoa, H.: Statistical Approach to Multipitch Analysis, PhD Thesis, The University of Toyo (2007). [4] Bregman, A. S.: Auditory scene analysis: The perceptual organization of sound, MIT press (1994). [5] Smaragdis, P. and Brown, J. C.: Non-negative matrix factorization for polyphonic music transcription, Proc. IEEE Worshop Applications Signal Process. Audio Acoust., IEEE, pp (2003). [6] No. 39 (2014). [7] No. 26 (2015). [8] Naamura, T., Shiata, K., Taamune, N. and Kameoa, H.: Harmonic-Temporal Factor Decomposition Incorporating Music Prior Information for Informed Monaural Source Separation, Proc. Int. Symposium Music Info. Retrieval, pp (2014). [9] [Online: 18, Apr. 2015], mirex/wii/mirex_home. [10] Smaragdis, P. and Mysore, G. J.: Separation by humming : User-guided sound extraction from monophonic mixtures, Proc. IEEE Worshop Applications Signal Process. Audio Acoust., IEEE, pp (2009). [11] Ozerov, A., Févotte, C., Blouet, R. and Durrieu, J. L.: Multichannel nonnegative tensor factorization with structured constraints for user-guided audio source separation, Proc. Int. Conf. Acoust. Speech Signal Process., IEEE, pp (2011). [12] Hennequin, R., David, B. and Badeau, R.: Score informed audio source separation using a parametric model of nonnegative spectrogram, Proc. Int. Conf. Acoust. Speech Signal Process., pp (2011). [13] Simseli, U. and Cemgil, A. T.: Score guided musical source separation using generalized coupled tensor factorization, Proc. Eur. Signal Process. Conf., IEEE, pp (2012). [14] F 0 Vol. SP , pp (2010). [15] Naano, M., Le Roux, J., Kameoa, H., Ono, N. and Sagayama, S.: Infinite-state spectrum model for music signal analysis, Proc. Int. Conf. Acoust. Speech Signal Process., pp (2011). [16] Raczyńsi, S. A., Ono, N. and Sagayama, S.: Multipitch analysis with harmonic nonnegative matrix approximation, Proc. Int. Conf. Music Info. Retrieval, pp (2007). [17] Vincent, E., Bertin, N. and Badeau, R.: Harmonic and inharmonic Nonnegative Matrix Factorization for Polyphonic Pitch transcription, Proc. Int. Conf. Acoust. Speech Signal c 1959 Information Processing Society of Japan 7

8 Process., pp (2008). [18] Yoshii, K. and Goto, M.: Infinite Latent Harmonic Allocation: A Nonparametric Bayesian Approach to Multipitch Analysis, Proc. Int. Soc. Music Info. Retrieval, pp (2010). [19] Hinton, G. E.: Training products of experts by minimizing contrastive divergence, Neural Comput., Vol. 14, No. 8, pp (2002). [20] Goto, M.: Development of the RWC Music Database, Proc. Int. Congress Acoust., pp. l (2004). [21] , (20. Nov. 2008). [22] [Online: 21, Apr. 2015], c 1959 Information Processing Society of Japan 8

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

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