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1 ,a,b (F0 (NMF (AR F0 (VB (MU. (Nonnegative Matrix Factorization: NMF [ 3] NMF [4,5] NMF (Multiplicative Update: MU NMF ( (F0 Umezono --, Tsuuba, Ibarai , Japan a.yoshii(ataist.go.jp b m.goto(ataist.go.jp (2 NMF F0 NMF F0 F0 (3 NMF NMF [ 3] (Infinite Composite Autoregressive Model: icarm ( [] c 202 Information Processing Society of Japan

2 I IJ IJ J I J IJ I J F0 (2 [7, 9] F0 [7,6] (3 [2] [] icarm (Variational Bayes: VB (MU NMF # # Kameoa [] IS - I AR J Badeau [2] IS H I MA J Durrieu [3] IS (H I - J Virtanen [4] KL - I - J Carabias-Orti [5] KL H I - J Heittola [6] KL - I - J [7] KL H+N I AR J Hennequin [8] Beta (0.5 - I ARMA J KL or IS H+N AR (HN- 2. NMF 2. NMF [4]NMF NMF 2.. NMF (KL [4] (IS [5] NMF [7] IS-NMF 3.2. KL-NMF IS KL-iCARM IS-iCARM 2..2 NMF [] AR [3 5] Badeau [2] (MA c 202 Information Processing Society of Japan 2

3 Heittola [6] [7] Hennequin [8] (ARMA MU 2..3 NMF Vincent [0] NMF Badeau [2] [7] Hennequin [9] Carabias-Orti [5] [5, 9] MU [7] AR 2.2 [8] NMF [9] 2 M,N I,J X m n Y m n θ i i φ j j W im i m A jm j m H n n i j Hoffman NMF NMF (GaP-NMF [2] [3] 3. (icarm 2IS AR [] KL 3.2. [7] [2] 3. 2M N M N X I J m, n, i, j X 3 W A H X or X 2 I,J i,j θ i φ j W im A jm H n ( W im A jm H n i m j m n i j 2 θ i φ j c 202 Information Processing Society of Japan 3

4 i j I J θ φ I + J + X p(θ, φ, H X; W, A W A p(θp(φp(h p(x θ, φ, H; W, A W A 3.2 KL IS icarm 3.2. X KL IS Y Y = Y Y = θ i φ j W im A jm H n ( X Y Y X Y X i j X X = X KL-iCARM X = X X Poisson(Y X Poisson( Y X X Poisson (Y (2 IS-iCARM X = X X N c (0,Y X N c (0, Y X 2 X 2 Exponential (Y ( θ φ θ φ θ φ ( α ( γ θ i Gamma I,α, φ j Gamma J,γ (4 I θ α ɛ>0 i j 2 W im A jm W im A jm 2 σ µ i 2µ i µ i 3 m W i A j θ i >ɛ I + I α θ I φ H H [20] H Gamma (β, β/d G n Gamma (β, βh n H n Gamma (β, βg n (5 β G n H n H n E prior [G n ]=H n E prior[h n ]=G n G n p(h n H n = Γ(2β (H n H n β 2Γ(β (H n +H n H 2β n W W i H 2H H W im = exp ( (m hμ i 2 2σ 2 (6 h= μ * i σ W Im = A i j x {x t } 2M t= *2 P P x t = a j px t p + s i t (7 p= s i {s i t} 2M t= i a j {a j 0,,aj p} T j a j 0 = * μ i [Hz] μ i = μ i /(r/2m [bins] r *2 2M m m c 202 Information Processing Society of Japan 4

5 w i {w i t }2M t= si {W im } 2M m= wi (DFT s i {X m }2M m= x DFT s i X m N c(0, Σ m Σ m = W ima jm A jm A jm = P p=0 aj pe 2π m 2M pi 2 = a T j U (8 ma j 2U m (P + (P + Toeplitz [U m ] pq =cos(2π m 2M (p q X m 2 W im A jm x a j { X m 2 } M m= {W ima jm } M m= IS *3 KL IS icarm A KL jm = a T j U or A IS jm = ma j a T j U (9 ma j KL-iCARM log p(x; W, A p(θ, φ, H X; W, A W A μσa (VB q(θ, φ, H = i q(θ i j q(φ j q(h n (0 n KL L *4 L log p(x; W, A E[log p(x θ, φ, H; W, A] + E[log p(θ] + E[log p(φ] + E[log p(h] E[log q(θ] E[log q(φ] E[log q(h] L ( q(θ exp(e q(φ,h [log p(x, θ, φ, H; W, A] q(φ exp(e q(θ,h [log p(x, θ, φ, H; W, A] q(h exp(e q(θ,φ [log p(x, θ, φ, H; W, A] (2 *3 (Linear Predictive Coding: LPC s i W im = *4 L W Aμσ a VB (MU [8, 9] MU L μ i L μ i = G μi F μi μ i μ i Fμ i G μi μ i 3.3. KL-iCARM KL-iCARM ( E[log p(x θ, φ, H; W, A] E[log p(x θ, φ, H; W, A] = log Poisson( X Y = [ X E log ] θ i φ j W im A jm H n E [ ] θ i φ j W im A jm H n log X! (3 log(x λ 0 λ = λ = {λ,,λ K } ( ( x log x =log λ ( x λ log (4 λ λ (3 E[log p(x θ, φ, H; W, A] X [ λ E log θ ] iφ j W im A jm H n λ E [ ] θ i φ j W im A jm H n log X! (5 λ = λ (5 log(x x (4,(5 θ, φ, H q(θ i = Gamma(a θ i,bθ i, q(φ j = Gamma(a φ j,bφ j q(h n = Gamma(a H n,b H n (6 a θ i = α I + j X λ b θ i = α + j E[φ jw im A jm H n ] a φ j = γ J + i X λ b φ j = γ + i E[θ iw im A jm H n ] a H n =2β + m X λ (7 b H n = βe[g n +G n+ ]+ m E[θ iφ j W im A jm ] c 202 Information Processing Society of Japan 5

6 (5 λ λ exp(e[log(θ i φ j W im A jm H n ] (8 λ m n i j θ i φ j W im A jm H n (8 θ i φ j W im A jm H n E[θ i φ j W im A jm H n ] θ E[θ i ]=exp(log(a θ i /bθ i exp(e[log θ i] = exp(ψ(a θ i /bθ i a θ i 3 NMF (MAP IS-iCARM IS-iCARM ( E[log p(x θ, φ, H; W, A] E[log p(x θ, φ, H; W, A] = log Exponential( X 2 Y = [ ] X 2 E θ iφ j W im A jm H n [ E log ] θ i φ j W im A jm H n (9 [2, 2] x η 0 η = η = {η,,η K } x = η x η x = η 2 (20 η η x (3 log(x ω log(x log(ω+ x (2 ω ω (9 E[log p(x θ, φ, H; W, A] X [ ] 2 η 2 E (22 θ i φ j W im A jm H n log(ξ + E [ ] θ i φ j W im A jm H n ξ η = η ξ 3 x y = exp(log(x y = exp( ψ ( x x K a+ x y = (2 x a = 2 K (2x a a a = a = 0 a = ψ(x K a (x (22 log(x x θ, φ, H log(x x log(xx x (GIG [2] q(θ i =GIG(a θ i,b θ i,c θ i, q(φ j = GIG(a φ j,bφ j,cφ j q(h n = GIG(a H n,bh n,ch n (23 E[φ jw ima jmh n] ξ a θ i = α I, bθ i = α + j c θ i = j X 2 η 2 E[ ] φ jw ima jmh n E[θ iw ima jmh n] ξ a φ j = γ J, bφ j = γ + i c φ j = i X 2 η 2 E[ ] θ iw ima jmh n a H n =2β, ch n = m X 2 η 2 E[ θ iφ jw ima jm ] b H n = βe[g n +G n+ ]+ m E[θ iφ jw ima jm] ξ (24 (22 η ξ η E[ θ iφ jw ima jmh n ] (25 ξ = E[θ iφ j W im A jm H n ] (26 (22 η 2 KL-iCARM λ m n X 2 (25 θ i φ j W im A jm H n E[θ i φ j W im A jm H n ] θ E[θ i ]= Ka+(2 bc c K a(2 bc E[ b θ i ] = Ka(2 bc c K a (2 bc b θ i E[θ i] E[ θ i ] 3 a θ i 0 c θ i (5 G q(g n = Gamma(a G n,bg n a G n =2β, b G n = βe[h n +H n ] (27 KL-iCARM c 202 Information Processing Society of Japan 6

7 3.3.3 KL- IS-iCARM W A μσ a (5 (22 log p(x; W,A μ i G μ i F μi μ i σ 2 G σ F 2 σ 2σ 2 a j G a j F aj a j F μi = ( jh h(mv F + hμ iv G exp (m hμi2 2σ 2 G μi = ( jh h(mv G + hμ iv F exp (m hμi2 F σ 2 G σ 2 = h V F (m hμ i 2 exp = h V G (m hμ i 2 exp 2σ ( 2 (m hμi2 2σ ( 2 (m hμi2 2σ 2 KL-iCARM V F = E[θ iφ j A jm H n ], F aj = i θ iφ j W im H n A 3 jm U m (28 V G = X λ W im G aj = i X λ A 2 jm U m (29 IS-iCARM V F = E[θ iφ j A jm H n ] ξ [ V G = X 2 ηe 2 ] θ iφ jwim 2 AjmHn F aj = E[θ iφ jw imh n] i ξ A 2 jm U m G aj = [ i X 2 η 2 E θ iφ jw imh n ]U m (30 a j G a j F aj a j a j 0 = a j 0 = A j 4. KL IS icarmkl-icarm IS-iCARM 4. [5,] MAPS [22] ENSTDCl [Hz] 6 [Hz] (STFT M = 024 N = 3000 MN X = max X 2 = I =88+,J = 0, α =, β = γ =0., H = 20, P =4,d = E emp [ X ]ore emp [ X 2 ] J =0 {μ i } 88 i= 88 n i j θ iφ j H n μ i 30 F 4.2 alb se2 4.9 D4, C#4, C4, A3, F#3 4 KL-iCARM 5 m n i E[Y i ]= j E[θ iφ j W im A jm H n ] X 5 IS-iCARM F0 IS Y X KL-iCARM IS-iCARM F % 35.% [5,] icarm 5. (icarm c 202 Information Processing Society of Japan 7

8 情報処理学会研究報告 観測データ X 再構成データ Y 観測データ X 再構成データ Y Y 中の調波成分 Y 中の雑音成分 Y 中の調波成分 Y 中の雑音成分 ソースの重み E[θ] 優位なフィルタ A ([db] ソースの重み E[θ] 優位なフィルタ A ([db] C#4 雑音 D4 F#3 A3 C4 必要なソースの個数が 過大評価されている 小さな重みを持つ 不要なソース フィルタの重み E[φ] フィルタの重み E[φ] 必要なフィルタの個数が 過大評価されている 非常に小さな重みを 持つ不要なフィルタ 図 4 KL-iCARM による音源分離結果 図 5 謝辞: 本研究の一部は JSPS 科研費 および JST OngaCREST プロジェクトの支援を受けた 参考文献 [] [2] [3] [4] [5] [6] [7] [8] [9] [0] [] H. Kameoa and K. Kashino. Composite autoregressive system for sparse source-filter representation of speech. ICASSP, pp , R. Badeau, V. Emiya, and B. David. Expectationmaximization algorithm for multi-pitch estimation and separation of overlapping harmonic spectra. ICASSP, pp , J.-L. Durrieu, G. Richard, B. David, and C. Fe votte. Source/Filter model for unsupervised main melody extraction from polyphonic audio signals. IEEE Trans. on ASLP, 8(3: , 200. T. Virtanen and A. Klapuri. Analysis of polyphonic audio using source-filter model and non-negative matrix factorization. NIPS Worshop, J. J. Carabias-Orti, T. Virtanen, P. Vera-Candeas, N. Ruiz-Reyes, and F. J. Can adas-quesada. Musical instrument sound multi-excitation model for non-negative spectrogram factorization. IEEE J. of Sel. Top. in Sig. Proc., 5(6:44 58, 20. T. Heittola, A. Klapuri, and T. Virtanen. Musical instrument recognition in polyphonic audio using sourcefilter model for sound separation. ISMIR, pp , 安良岡直希, 奥乃博. 調波 非調波 音色構造因子分解 による音響信号分析と音源分離インターフェースへの応 用. 情処研報, 202-MUS-94, pp. 8, 202. R. Hennequin, R. Badeau, and B. David. NMF with time-frequency activations to model nonstationary audio events. IEEE Trans. on ASLP, 9(4: , 20. R. Hennequin, R. Badeau, and B. David. Timedependent parametric and harmonic templates in nonnegative matrix factorization. DAFx, pp. 8, 200. E. Vincent, N. Bertin, and R. Badeau. Adaptive harmonic spectral decomposition for multiple pitch estimation. IEEE Trans. on ASLP, 8(3: , 200. N. Bertin, R. Badeau, and E. Vincent. Enforcing har- c 202 Information Processing Society of Japan [2] [3] [4] [5] [6] [7] [8] [9] [20] [2] [22] IS-iCARM による音源分離結果 monicity and smoothness in Bayesian non-negative matrix factorization applied to polyphonic music transcription. IEEE Trans. on ASLP, 8(3: , 200. M. Hoffman, D. Blei, and P. Coo. Bayesian nonparametric matrix factorization for recorded music. ICML, 200. M. Naano et al. Bayesian nonparametric spectrogram modeling based on infinite factorial infinite hidden Marov model. WASPAA, pp , 20. D. Lee and H. Seung. Algorithms for non-negative matrix factorization. NIPS, pp , C. Fe votte, N. Bertin, and J.-L. Durrieu. Nonnegative matrix factorization with the Itaura-Saito divergence: With application to music analysis. NECO, 2(3: , B. Niedermayer. Improving accuracy of polyphonic music-to-score alignment. ISMIR, pp , C. Fe votte and J. Idier. Algorithms for nonnegative matrix factorization with the beta-divergence. NECO, 23(9: , 20. P. Orbanz and Y. W. Teh. Bayesian nonparametric models. Encyclopedia of Machine Learning. Springer, 200. A. T. Cemgil. Bayesian inference for nonnegative matrix factorisation models. Computational Intelligence and Neuroscience, 2009:Article ID 78552, A. T. Cemgil and O. Dimen. Conjugate gamma Marov random fields for modelling nonstationary sources. ICA, 亀岡弘和, 後藤真孝, 嵯峨山茂樹. スペクトル制御エンベ ロープによる混合音中の周期および非周期成分の選択的 イコライザ. 情処研報, 2006-MUS-66, pp , V. Emiya, R. Badeau, and B. David. Multipitch estimation of piano sounds using a new probabilistic spectral smoothness principle. IEEE Trans. on ASLP, 8(6: , 200. 付 Gamma(x a, b = ba a bx e, x Γ(a 録 Exponential(x λ = x e λ λ a Poisson(x λ = λx λ (b/c 2 xa (bx+ c x e e, GIG(x a, b, c = x! 2Ka (2 bc 8

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