ブラインド音源分離における多変量複素Student's t 分布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,
“ブラインド音源分離における多変量複素Student’s t 分布に基づくランク制約付き空間共分散モデルの推定,”
IEICE Technical Report, EA2018-128, vol. 118, no. 495, pp.173–178, Nagasaki, March 2019.

Transcript

1. ϒϥΠϯυԻݯ෼཭ʹ͓͚Δ ଟมྔෳૉStudent’s t ෼෍ʹجͮ͘ ϥϯΫ੍໿෇͖ۭؒڞ෼ࢄϞσϧͷਪఆ ٱอ ༏ٍ 1ɼߴफ యݰ 1ɼ๺ଜ େ஍ 2ɼԐ౉ ༸ 1 1 ౦େ 2 ߳઒ߴઐ 2019 ೥ 3 ݄ 14 ೔
2. ɹ ຊݚڀͷର৅ͱ໨త 2 / 22 • എܠɿϒϥΠϯυԻݯ෼཭ ▶ ର৅ɿ͍͔ͭ͘ͷԻݯ͔ΒͷԻ͕౸དྷ ▶ ໨తɿ؍ଌࠞ߹Ի͔ΒݸʑͷԻݯ΁෼཭ ▶ ੍໿ɿԻڹతɾۭؒతಛ௃͸ະ஌ • ຊݚڀͷϑΥʔΧε ▶ ର৅ɿ֦ࢄੑԻݯதʹ 1 ͭͷ఺Իݯ͕ଘࡏ ▶ ໨తɿλʔήοτ఺ԻݯԻ੠ͷ෼཭ɾநग़ ▶ ຊݚڀɿ ෳૉ Student’s t ෼෍ʹΑΔϞσϧԽ ߴ଎ͳʢऩଋอূ෇ͷʣਪఆΞϧΰϦζϜ ⾼速
3. ɹ ຊݚڀͷର৅ͱ໨త 2 / 22 • എܠɿϒϥΠϯυԻݯ෼཭ ▶ ର৅ɿ͍͔ͭ͘ͷԻݯ͔ΒͷԻ͕౸དྷ ▶ ໨తɿ؍ଌࠞ߹Ի͔ΒݸʑͷԻݯ΁෼཭ ▶ ੍໿ɿԻڹతɾۭؒతಛ௃͸ະ஌ • ຊݚڀͷϑΥʔΧε ▶ ର৅ɿ֦ࢄੑԻݯதʹ 1 ͭͷ఺Իݯ͕ଘࡏ ▶ ໨తɿλʔήοτ఺ԻݯԻ੠ͷ෼཭ɾநग़ ▶ ຊݚڀɿ ෳૉ Student’s t ෼෍ʹΑΔϞσϧԽ ߴ଎ͳʢऩଋอূ෇ͷʣਪఆΞϧΰϦζϜ ⾼速
4. Background ଟνϟωϧඇෛ஋ߦྻҼࢠ෼ղ (MNMF) [Sawada+, 13] 3 / 22 • ֦ࢄੑͷԻݯ͕ϞσϧՄೳͳϒϥΠϯυԻݯ෼཭ख๏ • ύϥϝʔλ਺͕ଟ͘ɺਪఆʹ͕͔͔࣌ؒΔ
5. Background ϥϯΫ 1 ۭؒϞσϧʹجͮ͘ϒϥΠϯυԻݯ෼཭ 4 / 22 ：周波数インデクス ： 時間インデクス • ༏ܾఆ৚݅ԼͰɼप೾਺ྖҬʹ͓͚Δॠ࣌ࠞ߹Ծఆ ʢ⇔ ఺ԻݯԾఆɼϥϯΫ 1 ۭؒϞσϧʣ xij = Ai sij ↓ sij = A−1 i xij = Wi xij • 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) [D. Kitamura+, 16]
6. Conventional Method ϥϯΫ੍໿෇͖ۭؒڞ෼ࢄϞσϧ [ٱอଞ, 18] 5 / 22 • ֦ࢄੑࡶԻ͸શํҐΑΓ౸དྷʢ఺ԻݯͰͳ͍ʣ • ILRMA ͳͲͷख๏Ͱ͸໨తԻͱಉ͡ํҐ͔Β ౸དྷ͢Δ֦ࢄੑࡶԻͷ෼཭͕ࠔ೉ ʢਪఆ໨తԻʹࡶԻ͕࢒ཹ͢Δʣ • M − 1 ݸͷਪఆࡶԻͷਫ਼౓͸ඇৗʹߴ͍ [Takahashi, 09] (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
7. Conventional Method ֬཰Ϟσϧ 6 / 22 • ؍ଌ৴߸ xij Λ໨తԻ hij ͱ֦ࢄੑԻݯ uij ͷ࿨ͰϞσϧԽ • ෼ࢄڞ෼ࢄߦྻʹΑΔදݱ
8. Proposed Method ఏҊ๏ಈػɿଟมྔෳૉ Student’s t ෼෍ 7 / 22 • ILRMAɼMNMF Ͱͷ෼෍Λෳૉ Gauss ෼෍͔Βෳૉ Student’s t ෼෍ʹ֦ு → ҆ఆͰ֎Ε஋ʹؤ݈ͳਪఆΛՄೳʹʢt-ILRMA [Mogami+, 17], t-MNMF [K. Kitamura+, 16]ʣ • ࣗ༝౓ ν > 0 ΛมԽͤ͞Δ͜ͱͰ෼෍ͷܗঢ়ΛมԽͤ͞ΒΕΔ → ν Λ ∞ʢෳૉ Gauss ෼෍ʹରԠʣΑΓখ͘͢͞Δͱ੄໺͕େ͖͘ͳΔ • ϥϯΫ੍໿෇͖ۭؒڞ෼ࢄϞσϧ΁΋ޮՌ͕͋ΔՄೳੑ p(x; 0, Σ, ν) = 2M Γ( ν+2M 2 ) (νπ)M Γ(ν 2 ) det Σ × ( 1 + 2 ν xHΣ−1x ) − ν+2M 2 x ∼ Tν (0, Σ) ͷີ౓ؔ਺
9. Proposed Method ఏҊ๏ɿ֬཰Ϟσϧͱίετؔ਺ 8 / 22 • ؍ଌ৴߸ xij ͸ଟมྔෳૉ Student’s t ෼෍ Tν (0, R(x) ij ) ʹै͏ͱ͢Δ • ࠷దԽ͢΂͖ίετؔ਺ʢෛର਺໬౓ؔ਺ʣ f(Θ) = ∑ i,j [ ν + 2M 2 log ( 1 + 2 ν xH ij (R(x) ij )−1xij ) + (α + 1) log r(h) ij + β r(h) ij + log det R(x) ij ] + const.
10. Proposed Method ࠷దԽख๏ɿMMɾME ΞϧΰϦζϜ 9 / 22 • ௚઀తͳ࠷దԽͷࠔ೉ͳؔ਺ͷ࠷దԽʹ༻͍ΒΕΔ • Majorization-minimizationʢMMʣΞϧΰϦζϜ͸ MNMF ͳͲͰ ༻͍ΒΕ͍ͯΔ • Majorization-equalizationʢMEʣΞϧΰϦζϜ͸ MM ΞϧΰϦζϜΑΓ଎͍ऩଋΛ΋ͨΒ͢܏޲ [F´ evotte+, 11] ▶ NMF ʹ͓͍ͯద༻͞Ε͍ͯΔ ▶ ϑϧϥϯΫۭؒ૬ؔߦྻͷਪఆʹରͯ͠͸ద༻ྫ͕ͳ͍ • ෳૉ Student’s t ෼෍Ͱ͸ EM ΞϧΰϦζϜ͸ద༻ࠔ೉ • MMɾME ΞϧΰϦζϜΛ༻͍ͯ࠷దԽΛߦ͏ํ๏ΛఏҊ • ϑϧϥϯΫۭؒ૬ؔߦྻΛѻ͏ॳͷ ME ΞϧΰϦζϜಋग़
11. Proposed Method Majorization-minimization (MM) ΞϧΰϦζϜ 10 / 22 • ิ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 10 / 22 • ิ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 10 / 22 • ิ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-minimization (MM) ΞϧΰϦζϜ 10 / 22 • ิ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))
15. Proposed Method Majorization-minimization (MM) ΞϧΰϦζϜ 10 / 22 • ิ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))
16. Proposed Method Majorization-minimization (MM) ΞϧΰϦζϜ 10 / 22 • ิ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))
17. Proposed Method Majorization-equalization (ME) ΞϧΰϦζϜ 10 / 22 • ิ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))
18. Proposed Method ϥϯΫ੍໿෇͖ۭؒڞ෼ࢄϞσϧʹ͓͚Δิॿؔ਺ͷઃܭ 11 / 22 • ۭؒ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) = ν + 2M 2 log ( 1 + 2 ν xH ij (R(x) ij )−1xij ) + log det R(x) ij + (α + 1) log r(h) ij + β r(h) ij ิॿؔ਺ͷઃܭ͸༰қ
19. Proposed Method ϥϯΫ੍໿෇͖ۭؒڞ෼ࢄϞσϧʹ͓͚Δิॿؔ਺ͷઃܭ 11 / 22 • ۭؒ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) = ν + 2M 2 log ( 1 + 2 ν 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 ρij − 1) + (α + 1) r(h) ij ρij + β r(h) ij ≤ ≤
20. Proposed Method ϥϯΫ੍໿෇͖ۭؒڞ෼ࢄϞσϧʹ͓͚Δิॿؔ਺ͷઃܭ 11 / 22 • ۭؒ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) = ν + 2M 2 log ( 1 + 2 ν xH ij (R(x) ij )−1xij ) + log det R(x) ij + (α + 1) log r(h) ij + β r(h) ij 2 ν xH ij (R(x) ij )−1xij − τij 1 + τij + log(1 + τij ) tr(Ψ−1 ij (R(x) ij − Ψij )) + log det Ψij (α + 1)(log ρij − 1) + (α + 1) r(h) ij ρij + β r(h) ij ≤ ≤ ≤
21. Proposed Method Ұൠతͳߦྻʹର͢Δtr߲ͷෆ౳ࣜ 12 / 22 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 MNMF ʹ͓͚Δෆ౳ࣜ [Sawada+, 13]
22. Proposed Method Ұൠతͳߦྻʹର͢Δtr߲ͷෆ౳ࣜ 12 / 22 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 MNMF ʹ͓͚Δෆ౳ࣜ [Sawada+, 13]
23. Proposed Method ಘΒΕΔิॿؔ਺ͱิॿม਺ͷߋ৽ࣜ 13 / 22 • ҎԼͷิॿؔ਺ΛಘΔ f ≤ f++ := ν + 2M ν(1 + τij) ( 1 r(h) ij |ξH ij xij|2 + 1 r(u) ij xH ij ΦH ij (R(u) i )−1Φijxij ) + r(u) ij tr(Ψ−1 ij R(u) i ) + r(h) ij (a(h) i )HΨ−1 ij a(h) i + (α + 1) r(h) ij ρij + β r(h) ij + const. ʢͨͩ͠ R(u) i = R′(u) i + λibibH i ʣ • ౳߸੒ཱ৚݅Λݩʹิॿม਺Λߋ৽ʢ˜ r(h) ij ɿߋ৽લύϥϝʔλʣ 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 τij = 2 ν xH ij ( ˜ R(x) ij )−1xij , ρij = ˜ r(h) ij
24. Proposed Method ໨తม਺ͷ৐ࢉܕߋ৽ࣜ 14 / 22 • MM ΞϧΰϦζϜ ิॿม਺ߋ৽ޙɼิॿؔ਺Λ r(h) ij , r(u) ij , λi ʹؔͯ͠࠷খԽ ν → ∞ Ͱෳૉ Gauss ෼෍ͷ৔߹ͷߋ৽ଇͱҰக
25. Proposed Method ໨తม਺ͷ৐ࢉܕߋ৽ࣜ 15 / 22 • ME ΞϧΰϦζϜ ิॿม਺ߋ৽ޙɼิॿؔ਺ͷ஋Λม͑ͳ͍ r(h) ij , r(u) ij , λi Ͱߋ৽ ࠜ߸͕औΕɼߋ৽ͷมԽྔ͕େ͖͘ͳΔ → ߴ଎ͳߋ৽
26. Proposed Method ໨తม਺ͷ৐ࢉܕߋ৽ࣜ 15 / 22 • ME ΞϧΰϦζϜ ิॿม਺ߋ৽ޙɼิॿؔ਺ͷ஋Λม͑ͳ͍ r(h) ij , r(u) ij , λi Ͱߋ৽ MNMF ͳͲͷϑϧϥϯΫۭؒ૬ؔߦྻʹର͢Δ ME ΞϧΰϦ ζϜ͸ະใࠂ → ۭؒύϥϝʔλΛ 1 ࣗ༝౓ʹམͱ͠Մೳʹ
27. Experiments ࣮ݧ৚݅ 16 / 22 ϚΠΫ਺ 2, 3, 4 ໨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
28. Experiments ࣮ݧ݁Ռʢ2ϚΠΫʣ 17 / 22 0 5 10 15 20 25 30 Number of iterations 2.0 2.5 3.0 3.5 4.0 SDR improvement [dB] 2 mics ILRMA Gauss (MM, = ) Gauss (ME, = ) Student's t (MM, = 1) Student's t (ME, = 1) • 2 ϚΠΫͰ͸ɼෳૉ Student’s t ෼෍͕ෳૉ Gauss ෼෍ΑΓ ߴ଎ʹߋ৽Մೳ • ME ΞϧΰϦζϜ͸ MM ΞϧΰϦζϜΑΓߴ଎ʹߋ৽Մೳ
29. Experiments ࣮ݧ݁Ռʢ3ϚΠΫʣ 18 / 22 0 5 10 15 20 25 30 Number of iterations 4.5 5.0 5.5 6.0 6.5 7.0 SDR improvement [dB] 3 mics ILRMA Gauss (MM, = ) Gauss (ME, = ) Student's t (MM, = 1) Student's t (ME, = 1) • 3 ϚΠΫͷ৔߹ɼ෼෍ʹΑΔࠩ͸΄ͱΜͲݟΒΕͣ • ILRMA ͷੑೳ͕ྑ͘ɼ͔ͦ͜ΒߋͳΔ͕ࠩग़ͳ͍ͱߟ͑ΒΕΔ
30. Experiments ࣮ݧ݁Ռʢ4ϚΠΫʣ 19 / 22 0 5 10 15 20 25 30 Number of iterations 7 8 9 10 SDR improvement [dB] 4 mics ILRMA Gauss (MM, = ) Gauss (ME, = ) Student's t (MM, = 1) Student's t (ME, = 1) • 4 ϚΠΫͷ৔߹ɼ3 ϚΠΫͷ৔߹ͱಉ༷ʹɼ෼෍ʹΑΔࠩ͸ ΄ͱΜͲݟΒΕͣ
31. Experiments ࣮ݧ݁Ռʢ2ϚΠΫɼ࠶ܝʣ 20 / 22 0 5 10 15 20 25 30 Number of iterations 2.0 2.5 3.0 3.5 4.0 SDR improvement [dB] 2 mics ILRMA Gauss (MM, = ) Gauss (ME, = ) Student's t (MM, = 1) Student's t (ME, = 1) • 2 ϚΠΫͷ৔߹ɼILRMA ͷੑೳ͕ѱ͕ࠩ͘ग़Δͱߟ͑ΒΕΔ
32. Experiments ࣮ݧ݁Ռʢ2ϚΠΫɼࣗ༝౓ν ΛมԽʣ 21 / 22 • 2 ϚΠΫͷ৔߹ʹ༷ʑͳ ν Ͱ 10 ճ൓෮ • ν → 1 Ͱैདྷ๏ͱͷ͕ࠩ࠷΋ݦஶʹͳΔ
33. ·ͱΊ 22 / 22 • ʦ໨తʧ ํ޲ੑ໨తԻݯͱ֦ࢄੑԻݯͷ෼཭ • ʦखஈʧ ϥϯΫ੍໿෇͖ۭؒڞ෼ࢄϞσϧਪఆ๏ • ʦैདྷ๏ʧෳૉ Gauss ෼෍ͰϞσϧԽ • ʦಈػʧෳૉ Student’s t ෼෍Λಋೖͨ͠ଞैདྷख๏ͷଘࡏ • ʦ੒ՌʧMMɾME ΞϧΰϦζϜʹجͮ͘৐ࢉܕߋ৽ଇΛಋग़ • ʦ࣮ݧʧෳૉ Gauss ෼෍ʹجͮ͘ैདྷ๏ʹର͢Δ༏ҐੑΛ ɹɹɹɹ 2 ϚΠΫͷ৔߹ʹ֬ೝ
34. 23 / 22
35. Appendix ILRMAɼMNMFͱैདྷ๏ʢEMΞϧΰϦζϜʣ 24 / 22 0 100 200 Number of iterations 0 2 4 6 8 10 SDR improvement [dB] Babble noise ILRMA Original MNMF ILRMA+MNMF EM • ࡢ೥ळقԻڹֶձʹ͓͍ͯ ILRMA, MNMF<EM (in SDR) Λ֬ೝ • ຊ೥य़قԻڹֶձʹ͓͍ͯ EM<(MM,) ME Λ֬ೝ
36. Appendix ଟมྔͷMEΞϧΰϦζϜಋग़ͷࠔ೉ੑ 25 / 22 • 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೉
37. Appendix MEɾMMΞϧΰϦζϜಋग़ͷํ๏ 26 / 22 • ิॿؔ਺͸͍ͣΕͷ໨తม਺ x ʹରͯ͠΋ f+(x) = ax + b x + c ͷܗʹมܗͰ͖Δ • ࠷খԽˠඍ෼ͯ͠ 0 ͱ͓͘ • ஋Λม͑ͳ͍఺ˠೋ࣍ํఔࣜΛղ͘