乗算型更新式に基づくランク制約付き空間共分散モデルの推定

久保優騎, 高宗典玄, 北村大地．猿渡洋,
“乗算型更新式に基づくランク制約付き空間共分散モデルの推定,”
日本音響学会 2019年春季研究発表会講演論文集, 2-6-1, pp.245–248, Tokyo, March 2019.

Transcript

1. ৐ࢉܕߋ৽ࣜʹجͮ͘ ϥϯΫ੍໿෇͖ۭؒڞ෼ࢄϞσϧͷਪఆ ˑٱอ ༏ٍ 1ɼߴफ యݰ 1ɼ๺ଜ େ஍ 2ɼԐ౉ ༸ 1 1 ౦େ 2 ߳઒ߴઐ 2019 ೥ 3 ݄ 6 ೔
2. ɹ ຊݚڀͷର৅ͱ໨త 2 / 15 • എܠɿϒϥΠϯυԻݯ෼཭ ▶ ର৅ɿ͍͔ͭ͘ͷԻݯ͔ΒͷԻ͕౸དྷ ▶ ໨తɿ؍ଌࠞ߹Ի͔ΒݸʑͷԻݯ΁෼཭ ▶ ੍໿ɿԻڹతɾۭؒతಛ௃͸ະ஌ • ຊݚڀͷϑΥʔΧε ▶ ର৅ɿ֦ࢄੑԻݯதʹ 1 ͭͷ఺Իݯ͕ଘࡏ ▶ ໨తɿλʔήοτ఺ԻݯԻ੠ͷ෼཭ɾநग़ ▶ ຊݚڀɿΑΓߴ଎ͳʢऩଋอূ෇ͷʣ ɹɹɹɹਪఆΞϧΰϦζϜ ⾼速
3. ɹ ຊݚڀͷର৅ͱ໨త 2 / 15 • എܠɿϒϥΠϯυԻݯ෼཭ ▶ ର৅ɿ͍͔ͭ͘ͷԻݯ͔ΒͷԻ͕౸དྷ ▶ ໨తɿ؍ଌࠞ߹Ի͔ΒݸʑͷԻݯ΁෼཭ ▶ ੍໿ɿԻڹతɾۭؒతಛ௃͸ະ஌ • ຊݚڀͷϑΥʔΧε ▶ ର৅ɿ֦ࢄੑԻݯதʹ 1 ͭͷ఺Իݯ͕ଘࡏ ▶ ໨తɿλʔήοτ఺ԻݯԻ੠ͷ෼཭ɾநग़ ▶ ຊݚڀɿΑΓߴ଎ͳʢऩଋอূ෇ͷʣ ɹɹɹɹਪఆΞϧΰϦζϜ ⾼速
4. ɹ ϥϯΫ 1 ۭؒϞσϧʹجͮ͘ϒϥΠϯυԻݯ෼཭ 3 / 15 ：周波数インデクス ： 時間インデクス • प೾਺ྖҬʹ͓͚Δॠ࣌ࠞ߹Ծఆ ʢ⇔ ఺ԻݯԾఆɼϥϯΫ 1 ۭؒϞσϧʣ xij = Ai sij • ਪఆ৴߸ yij = Wi xij ͷ֤੒෼͕ಠཱʹͳΔΑ͏෼཭ߦྻ Wi Λਪఆ ▶ प೾਺ྖҬಠཱ੒෼෼ੳ (FDICA) [Smaragdis, 98], [Saruwatari+, 06] ▶ ಠཱϕΫτϧ෼ੳ (IVA) [Hiroe, 06], [Kim+, 06] ▶ ಠཱ௿ϥϯΫߦྻ෼ੳ (ILRMA) [Kitamura+, 16]
5. Conventional Method ϥϯΫ੍໿෇͖ۭؒڞ෼ࢄϞσϧ [ٱอଞ, 18] 4 / 15 • ֦ࢄੑࡶԻ͸શํҐΑΓ౸དྷʢ఺ԻݯͰͳ͍ʣ • ILRMA ͳͲͷख๏Ͱ͸໨తԻͱಉ͡ํҐ͔Β ౸དྷ͢Δ֦ࢄੑࡶԻͷ෼཭͕ࠔ೉ ʢਪఆ໨తԻʹࡶԻ͕࢒ཹ͢Δʣ • M − 1 ݸͷਪఆࡶԻͷਫ਼౓͸ඇৗʹߴ͍ (M ͸ϚΠΫ਺ɾԻݯ਺) • Ի੠ͷํҐɾࡶԻͷϥϯΫ M − 1 ͷۭؒ૬ؔߦྻ͸ਖ਼֬ʹ෼͔Δ • ϥϯΫ੍໿෇͖ۭؒڞ෼ࢄϞσϧɿ ILRMA Λ༻͍ͯҰ෦ͷۭؒ৘ใΛਪఆͨ͠ޙɼ͚ܽͨϥϯΫ 1 ੒෼Λ ਪఆ͠ɼଟνϟωϧ Wiener ϑΟϧλΛ༻͍ͯ࢒ཹࡶԻΛ཈ѹ 空間的に分離困難
6. Conventional Method ֬཰Ϟσϧͱύϥϝʔλਪఆ 5 / 15 • ؍ଌ৴߸ xij Λ໨తԻ hij ͱ֦ࢄੑԻݯ uij ͷ࿨ͰϞσϧԽ • ύϥϝʔλਪఆɿEM ΞϧΰϦζϜ E-step M-step
7. Proposed Method ಈػɿਪఆΞϧΰϦζϜͷมߋʹΑΔߴ଎Խ 6 / 15 • Majorization-minimizationʢMMʣΞϧΰϦζϜ͕ ϑϧϥϯΫۭؒ૬ؔߦྻΛѻ͏ϞσϧͰ༻͍ΒΕ͍ͯΔ ▶ EM ΞϧΰϦζϜ͸ MM ΞϧΰϦζϜͷҰछ ▶ ิॿؔ਺Λ EM Ͱ༻͍ΒΕͨ΋ͷ͔Βมߋ͢Δͱ৐ࢉܕߋ৽ଇ ͕ಘΒΕɼߴ଎Խ͕ݟࠐΊΔ [Sawada+, 13] • Majorization-equalizationʢMEʣΞϧΰϦζϜ͕ MM ΞϧΰϦζϜΑΓ଎͍ऩଋΛ΋ͨΒ͢܏޲ [F´ evotte, 11] ▶ ϑϧϥϯΫۭؒ૬ؔߦྻʹରͯ͠͸ద༻ྫ͕ͳ͍ • ϥϯΫ੍໿෇͖ۭؒڞ෼ࢄϞσϧʹ͓͍ͯɼMMɾME Ξϧΰ ϦζϜͷಋग़ʹΑΓߋ৽Λߴ଎Խ • ϑϧϥϯΫۭؒ૬ؔߦྻΛѻ͏ॳͷ ME ΞϧΰϦζϜಋग़
8. Proposed Method Majorization-minimization (MM) ΞϧΰϦζϜ 7 / 15 • ิॿม਺ Ω ͱิॿؔ਺ f+ ͸࣍Λຬͨ͢ • Θ ͱ Ω ͷަޓ࠷దԽΛ܁Γฦ͢ f(Θ) ≤ f+(Θ, Ω) (∀Θ,∀ Ω) f(Θ) = min Ω f+(Θ, Ω) (∀Θ) Ω(n+1) ← arg min Ω f+(Θ(n), Ω) Θ(n+1) ← arg min Θ f+(Θ, Ω(n+1))
9. Proposed Method Majorization-minimization (MM) ΞϧΰϦζϜ 7 / 15 • ิॿม਺ Ω ͱิॿؔ਺ f+ ͸࣍Λຬͨ͢ • Θ ͱ Ω ͷަޓ࠷దԽΛ܁Γฦ͢ f(Θ) ≤ f+(Θ, Ω) (∀Θ,∀ Ω) f(Θ) = min Ω f+(Θ, Ω) (∀Θ) Ω(n+1) ← arg min Ω f+(Θ(n), Ω) Θ(n+1) ← arg min Θ f+(Θ, Ω(n+1))
10. Proposed Method Majorization-minimization (MM) ΞϧΰϦζϜ 7 / 15 • ิॿม਺ Ω ͱิॿؔ਺ f+ ͸࣍Λຬͨ͢ • Θ ͱ Ω ͷަޓ࠷దԽΛ܁Γฦ͢ f(Θ) ≤ f+(Θ, Ω) (∀Θ,∀ Ω) f(Θ) = min Ω f+(Θ, Ω) (∀Θ) Ω(n+1) ← arg min Ω f+(Θ(n), Ω) Θ(n+1) ← arg min Θ f+(Θ, Ω(n+1))
11. Proposed Method Majorization-minimization (MM) ΞϧΰϦζϜ 7 / 15 • ิॿม਺ Ω ͱิॿؔ਺ f+ ͸࣍Λຬͨ͢ • Θ ͱ Ω ͷަޓ࠷దԽΛ܁Γฦ͢ f(Θ) ≤ f+(Θ, Ω) (∀Θ,∀ Ω) f(Θ) = min Ω f+(Θ, Ω) (∀Θ) Ω(n+1) ← arg min Ω f+(Θ(n), Ω) Θ(n+1) ← arg min Θ f+(Θ, Ω(n+1))
12. Proposed Method Majorization-minimization (MM) ΞϧΰϦζϜ 7 / 15 • ิॿม਺ Ω ͱิॿؔ਺ f+ ͸࣍Λຬͨ͢ • Θ ͱ Ω ͷަޓ࠷దԽΛ܁Γฦ͢ f(Θ) ≤ f+(Θ, Ω) (∀Θ,∀ Ω) f(Θ) = min Ω f+(Θ, Ω) (∀Θ) Ω(n+1) ← arg min Ω f+(Θ(n), Ω) Θ(n+1) ← arg min Θ f+(Θ, Ω(n+1))
13. Proposed Method Majorization-minimization (MM) ΞϧΰϦζϜ 7 / 15 • ิॿม਺ Ω ͱิॿؔ਺ f+ ͸࣍Λຬͨ͢ • Θ ͱ Ω ͷަޓ࠷దԽΛ܁Γฦ͢ f(Θ) ≤ f+(Θ, Ω) (∀Θ,∀ Ω) f(Θ) = min Ω f+(Θ, Ω) (∀Θ) Ω(n+1) ← arg min Ω f+(Θ(n), Ω) Θ(n+1) ← arg min Θ f+(Θ, Ω(n+1))
14. Proposed Method Majorization-equalization (ME) ΞϧΰϦζϜ 7 / 15 • ิॿม਺ Ω ͱิॿؔ਺ f+ ͸࣍Λຬͨ͢ • Θ ͱ Ω ͷަޓ࠷దԽΛ܁Γฦ͢ • ME ΞϧΰϦζϜ͸ MM ΞϧΰϦζϜʹ ൺ΂ͯߴ଎Ͱ͋Δ܏޲ [F´ evotte, 11] • ϑϧϥϯΫۭؒ૬ؔߦྻΛѻ͏ Ϟσϧʹରͯ͠͸ద༻ྫແ͠ → ଟมྔͷ৔߹ಋग़͕ࠔ೉ͳͨΊ f(Θ) ≤ f+(Θ, Ω) (∀Θ,∀ Ω) f(Θ) = min Ω f+(Θ, Ω) (∀Θ) Ω(n+1) ← arg min Ω f+(Θ(n), Ω) Θ(n+1) ← ˆ Θ s.t. f+(ˆ Θ, Ω(n+1)) = f+(Θ, Ω(n+1))
15. Proposed Method ϥϯΫ੍໿෇͖ۭؒڞ෼ࢄϞσϧʹ͓͚Δิॿؔ਺ͷઃܭ 8 / 15 • ۭؒ૬ؔߦྻͱ֤࣌ؒप೾਺ϑϨʔϜͰͷίετؔ਺ R(x) ij = r(h) ij a(h) i (a(h) i )H + r(u) ij R(u) i , R(u) i = R′(u) i + λibibH i f(r(h) ij , r(u) ij , λi) = xH ij (R(x) ij )−1xij + log det R(x) ij + (α + 1) log r(h) ij + β r(h) ij ิॿؔ਺ͷઃܭ͸༰қ
16. Proposed Method ϥϯΫ੍໿෇͖ۭؒڞ෼ࢄϞσϧʹ͓͚Δิॿؔ਺ͷઃܭ 8 / 15 • ۭؒ૬ؔߦྻͱ֤࣌ؒप೾਺ϑϨʔϜͰͷίετؔ਺ R(x) ij = r(h) ij a(h) i (a(h) i )H + r(u) ij R(u) i , R(u) i = R′(u) i + λibibH i f(r(h) ij , r(u) ij , λi) = xH ij (R(x) ij )−1xij + log det R(x) ij + (α + 1) log r(h) ij + β r(h) ij tr(Ψ−1 ij (R(x) ij − Ψij )) + log det Ψij (α + 1)(log ˜ r(h) ij − 1) + (α + 1) r(h) ij ˜ r(h) ij + β r(h) ij ≤ ≤
17. Proposed Method Ұൠతͳߦྻʹର͢Δtr߲ͷෆ౳ࣜ 9 / 15 Λ ∈ CM×M : ൒ਖ਼ఆ஋Τϧϛʔτ ɹ Rn:ਖ਼ఆ஋Τϧϛʔτɼ ∑ n Φn = I tr(( ∑ n Rn)−1Λ) ≤ ∑ n tr(ΦH n R−1 n ΦnΛ) xH ij (R(x) ij )−1xij = tr((R(x) ij )−1xijxH ij ) R(x) ij = r(h) ij a(h) i (a(h) i )H + r(u) ij R(u) i ଟνϟωϧ NMFʢMNMFʣʹ͓͚Δෆ౳ࣜ [Sawada+, 13]
18. Proposed Method Ұൠతͳߦྻʹର͢Δtr߲ͷෆ౳ࣜ 9 / 15 Λ ∈ CM×M : ൒ਖ਼ఆ஋Τϧϛʔτ ɹ Rn:ਖ਼ఆ஋Τϧϛʔτɼ ∑ n Φn = I tr(( ∑ n Rn)−1Λ) ≤ ∑ n tr(ΦH n R−1 n ΦnΛ) ɹ Rn ɿ൒ਖ਼ఆ஋Τϧϛʔτɼ ∑ n Rn ɿਖ਼ଇɼ ∑ n Φn = PɼP ɿ ImΛ ΁ͷࣹӨߦྻɼKerΦn = KerΛɼImΦn = ImRn tr(( ∑ n Rn)−1Λ) ≤ ∑ n tr(ΦH n R+ n ΦnΛ) Rn ͷϥϯΫʹؔ͢ΔҰൠԽ ଟνϟωϧ NMFʢMNMFʣʹ͓͚Δෆ౳ࣜ [Sawada+, 13]
19. Proposed Method ಘΒΕΔิॿؔ਺ͱิॿม਺ͷߋ৽ࣜ 10 / 15 • ߋ৽લͷύϥϝʔλΛ ˜ r(h) ij ͳͲͱ͢Δ f ≤ f+ := |ξH ij xij|2 + β r(h) ij + r(h) ij ( (a(h) i )HΨ−1 ij a(h) i + α + 1 ˜ r(h) ij ) + 1 r(u) ij xH ij ΦH ij (R(u) i )−1Ψijxij + r(u) ij tr(Ψ−1 ij R(u) i ) + const. (ͨͩ͠R(u) i = R′(u) i + λibibH i ) • ౳߸੒ཱ৚݅Λݩʹิॿม਺Λߋ৽ Ψij = ˜ R(x) ij ξij = ˜ r(h) ij ( ˜ R(x) ij )−1a(h) i Φij = ˜ r(u) ij ˜ R(u) i ( ˜ R(x) ij )−1
20. Proposed Method ໨తม਺ͷ৐ࢉܕߋ৽ࣜ 11 / 15 • MM ΞϧΰϦζϜ ิॿม਺ߋ৽ޙɼิॿؔ਺Λ r(h) ij , r(u) ij , λi ʹؔͯ͠࠷খԽ r(h) ij = ˜ r(h) ij |(a(h) i )H(R(x) ij )−1xij |2 + β (˜ r(h) ij )2 (a(h) i )H(R(x) ij )−1a(h) i + α+1 ˜ r(h) ij r(u) ij = ˜ r(u) ij xH ij (R(x) ij )−1R(u) i (R(x) ij )−1xij tr((R(x) ij )−1R(u) i ) λi = ˜ λi ∑ j r(u) ij |bH i (R(x) ij )−1xij |2 ∑ j r(u) ij bH i (R(x) ij )−1bi
21. Proposed Method ໨తม਺ͷ৐ࢉܕߋ৽ࣜ 12 / 15 • ME ΞϧΰϦζϜ ิॿม਺ߋ৽ޙɼิॿؔ਺ͷ஋Λม͑ͳ͍ r(h) ij , r(u) ij , λi Ͱߋ৽ r(h) ij = ˜ r(h) ij |(a(h) i )H(R(x) ij )−1xij |2 + β (˜ r(h) ij )2 (a(h) i )H(R(x) ij )−1a(h) i + α+1 ˜ r(h) ij r(u) ij = ˜ r(u) ij xH ij (R(x) ij )−1R(u) i (R(x) ij )−1xij tr((R(x) ij )−1R(u) i ) λi = ˜ λi ∑ j r(u) ij |bH i (R(x) ij )−1xij |2 ∑ j r(u) ij bH i (R(x) ij )−1bi
22. Proposed Method ໨తม਺ͷ৐ࢉܕߋ৽ࣜ 12 / 15 • ME ΞϧΰϦζϜ ิॿม਺ߋ৽ޙɼิॿؔ਺ͷ஋Λม͑ͳ͍ r(h) ij , r(u) ij , λi Ͱߋ৽ r(h) ij = ˜ r(h) ij |(a(h) i )H(R(x) ij )−1xij |2 + β (˜ r(h) ij )2 (a(h) i )H(R(x) ij )−1a(h) i + α+1 ˜ r(h) ij r(u) ij = ˜ r(u) ij xH ij (R(x) ij )−1R(u) i (R(x) ij )−1xij tr((R(x) ij )−1R(u) i ) λi = ˜ λi ∑ j r(u) ij |bH i (R(x) ij )−1xij |2 ∑ j r(u) ij bH i (R(x) ij )−1bi • MNMF ͳͲͷϑϧϥϯΫۭؒ૬ؔߦྻʹର͢Δ ME ΞϧΰϦ ζϜ͸ະใࠂˠۭؒύϥϝʔλΛ 1 ࣗ༝౓ʹམͱ͠Մೳʹ
23. Experiments ࣮ݧ৚݅ 13 / 15 ໨తԻ੠৴߸ JNAS ΫϦʔϯԻݯσʔλϕʔεͷԻݯ (16 kHz) ࡶԻ৴߸ ަ௨ࡶԻ (DEMAND) ΠϯύϧεԠ౴ ࢒ڹ 200 ms ؀ڥԼͰऩ࿥ Ի੠ͱࡶԻͷ SNR 0 dB ૭௕ (FFT ௕) 1024 ఺ (64 ms ૬౰) γϑτ௕ 512 ఺ ILRMA ͷ൓෮ճ਺ 50 ධՁࢦඪ source-to-distortion ratio (SDR) վળྔ 6.45 cm 10° 1.5 m 1.0 m Target speech Noise sources Impulse response T60 = 200 ms
24. Experiments ࣮ݧ݁Ռ 14 / 15 0 10 20 30 Number of iterations −2 −1 0 1 2 Log-likelihood ×106 0 10 20 30 Number of iterations 7.9 8.2 8.5 8.8 9.1 9.4 9.7 10.0 SDR improvement [dB] EM MM ME EM MM ME • ಛʹ ME ͕ɼ଎͍໬౓ͷऩଋɾߴ͍ SDR վળΛࣔͨ͠
25. ·ͱΊ 15 / 15 • ʦ໨తʧ ํ޲ੑ໨తԻݯͱ֦ࢄੑԻݯͷ෼཭ • ʦखஈʧ ϥϯΫ੍໿෇͖ۭؒڞ෼ࢄϞσϧਪఆ๏ • ʦैདྷ๏ʧEM ΞϧΰϦζϜͰύϥϝʔλਪఆ • ʦಈػʧ MNMFɿิॿؔ਺๏ʹجͮ͘৐ࢉܕߋ৽ଇ͕༗ޮ • ʦ੒Ռ 1ʧϥϯΫ੍໿Λߟྀͨ͠ෆ౳ࣜΛಋग़ • ʦ੒Ռ 2ʧMMɾME ΞϧΰϦζϜʹجͮ͘৐ࢉܕߋ৽ଇΛಋग़ ▶ ϑϧϥϯΫۭؒ૬ؔߦྻΛѻ͏ϞσϧͰ͸ ME ͸ॳͷద༻ • ʦ࣮ݧʧ EM ΞϧΰϦζϜʹର͢Δ༏ҐੑΛ֬ೝ
26. 16 / 15
27. Appendix ILRMAɼMNMFͱैདྷ๏ʢEMΞϧΰϦζϜʣ 17 / 15 0 100 200 Number of iterations 0 2 4 6 8 10 SDR improvement [dB] Babble noise ILRMA Original MNMF ILRMA+MNMF EM • ࡢ೥Իڹֶձʹ͓͍ͯ ILRMA, MNMF<EM (in SDR) Λ֬ೝ • ຊൃදͰ͸ EM<(MM,) ME Λ ֬ೝ
28. Appendix ଟมྔͷMEΞϧΰϦζϜಋग़ͷࠔ೉ੑ 18 / 15 • ଟมྔͷϞσϧͰ͸ɼۭؒ૬ؔߦྻ R ∈ CM×M ʹରͯ͠ิॿ ؔ਺͸ f+(R) = tr(AR−1) + tr(BR) + const. ͱॻ͘͜ͱ͕Ͱ͖Δ • ͜ΕΛ࠷খԽ͢Δ఺ʢMM ΞϧΰϦζϜʣͷٻղ͸୅਺ Riccati ํఔࣜʹؼண [Sawada+,13] • ҰํͰ f+ Λม͑ͳ͍఺ʢME ΞϧΰϦζϜʣ͸ແ਺ʹଘࡏ͠ɼ ͔ͭͦͷΑ͏ͳ఺ͷू߹ΛٻΊΔ͜ͱ΋ࠔ೉
29. Appendix MEɾMMΞϧΰϦζϜಋग़ͷํ๏ 19 / 15 • ิॿؔ਺͸͍ͣΕͷ໨తม਺ x ʹରͯ͠΋ f+(x) = ax + b x + c ͷܗʹมܗͰ͖Δ • ࠷খԽˠඍ෼ͯ͠ 0 ͱ͓͘ • ஋Λม͑ͳ͍఺ˠೋ࣍ํఔࣜΛղ͘