†1 †1 (Infinite Latent Harmonic Allocation; iLHA) iLHA
Infinite Latent Harmonic Allocation
for Multipitch Analysis
K AZUYOSHI Y OSHII †1 and M ASATAKA G OTO†1
This paper presents a method called the infinite latent harmonic allocation (iLHA) for detecting multiple fundamental frequencies in polyphonic audio sig- nals. We regard amplitude spectra as histograms of observed-frequency counts and assume each frequency to be generated from one of the harmonic partials contained in a sound source. To formulate conventional probabilistic models, however, we have to specify the numbers of sources and partials, which are un- known in advance.  To avoid this problem, we propose a nonparametric Bayesian model that allows the infinite numbers of sources and partials in theory. More specifically, we formulated nested infinite Gaussian mixture models using hi- erarchical and generalized Dirichlet processes and derived a training method based on collapsed variational Bayes. Our experiments showed that fully auto- mated iLHA performed as well as optimally tuned conventional methods. ↑1
(AIST)
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100%
1 IPSJ SIG Technical Report Vol.2010-MUS-86 No.11 2010/7/29 4 2 3 LHA 5 iLHA 7 6 2. 1)–10) (Infinite Latent Harmonic Allocation; 2.1 D
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f M [cent] K X d Nd ZるXdn Zdn Zdnkm iLHA) 1) 2)
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x dn (1 ≤ n ≤ Nd) X d k (1 ≤ k ≤ K) 2 5 2) 1)
IPSJ SIG Technical Report HTC (NMF) fc = 1200log2(fh/(440 12 -5)) fc[cent] fん[Hz]物2010 Information Processing Society of Japan 3. iLHA 3.1 1) 2) k (1 ≤ k ≤ K) 1 M æ[cent]*¹ M (1) M k (x) = Then m = 1 Λ k P k k Tkm (1 ≤ m ≤ M) 1 0m = [1200 log 2 m] m M K 2 d 3) æ[cent] (2) (MAP) k: MAP d π dk 3.2 t 1) 3.3 iLHA K M 9) ★1 3 IPSJ SIG Technical Report 4. iLHA LHA PreFEst
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q*(Z) å exp (E ¥,7,µ,A [log p(X,Z,7,7,µ,A)])
(10) (11) 4.3.1 VB-E Z log q*(Z) = E π,τ,μ,Λ [log p(X, Z, π,τ,μ,Λ)] + const. = Ep,A [logp(X|Z, p, A)] + Ep,r [logp(Z|π, τ)] + const. Zdnkm log βdnkm + const. dnkm X (MCMC) MCMC q(Z, π, τ, μ, Λ) p(Z, π, τ, μ, Λ| X) p(X, Z, π, τ, μ, Λ) = p(X|Z, μ, Λ)p(Z|π, τ)p(π)p(τ)p(μ, Λ) Z
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k = 1 Go G d k 1 E P 3 (27) (28) (29) α < 1
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5 Go = (17) (18) δ κ 8 IPSJ SIG Technical Report Go D DP (6) V π d (Stick-Breaking Construction) 12) k — 1 ν k = ν k k' = 1 7 DP ν ~ GEM(γ) V π d 5.2 M k t 2 m — 1 Tkm = Tkm m' = 1 β
ァ～ GEM(βλ1,βλ2)
DP (HDP) 12) DP V d d V DP DP
V GEM(Griffiths-Engen-McCloskey) HDP DP DP
DP (7) DP (31) 5.3
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p(α) = Gam(α|α, bα) p(β) = Gam(ββαβ, bβ)
(33) b{α,β,γ} α{α,β,γ} u1, u2 5  p(X, Z, π, π, π, Λ, α, β, γ, λ, ν) = p(X|Z, µ, A)p(µ, A)p(Z|π, ¯)p(π|α, ¯)p(¯|β, λ)p(α)p(β)p(τ)p(λ)p(σ|γ) ρ(π|α, ν)
Beta( ī km | βλ) ρ( π | β, λ ) = km HDP (CVB) 13) VB (9) CVB CVB Go V DP d (6) (35) (34) p(¯)√|γ) = 5.4 VB 6物2010 Information Processing Society of Japan (30) DP (32) DP IPSJ SIG Technical Report
π,ﾃ, μ, Λ
bzk, W zk, Czk B(W zk , Czk ) (36)
13) Sdk
(16) Vdnkm B(Wo,co)
p(X, Z, η, ξ, s, t, α, β, γ, λ, ν) = p(Z, η, ξ, s, t(α, β, λ, ν)p(α)p(β)p(γ)p(λ)p(α)ρ(α)ρ(α) (36) CVB
(41) k, m q( Zdnkm ) = 0 K M
(42) q*( zdn ) ∞ exp ( E z-dn,α,β,γ,λ,α [log p( X, Z,α,β,γ,λ,α)[ ] q* (α, β, γ, λ) ∞ exp (E z,η,η,ε,τ,ϊ [log p(X, Z,η,ξ, s, t,α,β, γ, λ, ῦ)]) q*(¯) << exp (Ez,n,s,ξ,t,α,β,τ,λ [logp(X, Z,η,ξ, s,t,α,β,τ,λ,α)]) q* (η,ξ, s, t|Z) ∞ exp (Ε z,α,β,γ,λ,α [log p(X, Z, η,ξ, s, t, α,β,γ,λ,α)]) ¬dn d, n (43) (44) (45) 5.5 (42), (43), (44) (45) 5.5.1 CVB-E Zdnkm = 1
A (23) (24) Zdnkm.
7物2010 Information Processing Society of Japan G[x] V
IPSJ SIG Technical Report HTC iLHA LHA物2010 Information Processing Society of Japan 6. iLHA 16) 8 iLHA FO F M d FO Na k 3) FO 6.2 2 iLHA HTC HTC PreFEst n.k>m = 0 n.k>m > 0 13) LHA [sdk] E[tkm1] [ t km2] FO μ, Λ MAP FO CVB-E PreFEst HTC q(Z) VB-M iLHA 4 8
7.
(iLHA)
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JST CREST
20800084
(NTT)
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