ࣾஂ๏ਓ ిࢠ৘ใ௨৴ֶձ
THE INSTITUTE OF ELECTRONICS,
INFORMATION AND COMMUNICATION ENGINEERS৴ֶٕใTECHNICAL REPORT OF IEICE.

ʦϙελʔߨԋʧϚϧννϟωϧԻ‫ݯ‬෼཭ͷͨΊͷ௿ϥϯΫԻ‫ݯ‬Ϟσϧͱεύʔεॏ৞աఔʹ‫ͮ͘ج‬ωετ‫ܕ‬ϕΠζࠞ߹ɾҼࢠϞσϧ൘૔ ޫ༎†ࡔ౦ٓত†தଜ ӫଠ†

ࢳࢁ ࠀण†٢Ҫ࿨Ղ†Տ‫ ݪ‬ୡ໵†† ‫౎ژ‬େֶ େֶӃ৘ใֶ‫ڀݚ‬Պ
E-mail: †{itakura,bando,enakamura,itoyama,yoshii,kawahara}@sap.ist.i.kyoto-u.ac.jp͋Β·͠ຊߘͰ͸ɼ௿ϥϯΫԻ‫ݯ‬Ϟσϧͱεύʔεॏ৞աఔʹ‫ͮ͘ج‬ωετ‫ܕ‬ϕΠζࠞ߹ɾҼࢠϞσϧΛ༻͍ͨϚϧννϟωϧԻ‫ݯ‬෼཭๏ʹ͍ͭͯड़΂ΔɽैདྷͷԻ‫ݯ‬෼཭Ͱ͸ɼԻ‫ݯ‬Ϟσϧʹରͯ͠௿ϥϯΫੑΛԾఆ͠ҼࢠϞσϧΛ༻͍ͨϞσϧԽΛߦ͏ඇෛ஋ߦྻҼࢠ෼ղ΍ॏ৞աఔʹ͓͍ͯԻ‫ݯ‬ͷεύʔεੑΛԾఆࠞ͠߹Ϟσϧʹ‫ͮج‬͍ͨ࣌ؒप೾਺ΫϥελϦϯάʹΑΔԻ‫ݯ‬෼཭๏ͳͲ͕͋ͬͨɽఏҊ๏Ͱ͸͜ͷԻ‫ݯ‬Ϟσϧͱॏ৞աఔΛ౷߹ͨ͠ϞσϧԽʹΑΓԻ‫ݯ‬෼཭Λߦ͏ɽ·ͨɼҼࢠϞσϧͱࠞ߹Ϟσϧͷؔ܎ੑʹண໨͠ɼԻ‫ݯ‬Ϟσϧͱॏ৞աఔͷͦΕͧΕʹର͠ҼࢠϞσϧͱࠞ߹ϞσϧʹΑΔϞσϧԽΛߦ͏͜ͱͰෳ਺ͷ෼཭๏ΛఏҊ͢ΔɽΩʔϫʔυ ϚϧννϟωϧԻ‫ݯ‬෼཭ɼ࣌ؒप೾਺ΫϥελϦϯάɼજࡏతσΟϦΫϨ഑෼๏ɼඇෛ஋ߦྻҼࢠ෼ղ

1.

͸͡Ίʹ

·ͨɼ୯νϟωϧԻ‫ݯ‬෼཭ͷͨΊͷख๏ͱͯ͠ඇෛ஋ߦྻҼࢠ෼ղ (NMF) [11] ͕޿͘࢖ΘΕ͍ͯΔɽ୯νϟωϧԻ‫ݯ‬෼཭ϚΠΫϩϗϯΞϨΠΛ༻͍ͨϚϧννϟωϧԻ‫ݯ‬෼཭ʹ͓Ͱ͸ϚΠΫؒͷҐ૬ࠩͳͲͷۭؒతͳ৘ใΛ༻͍Δ͜ͱ͕Ͱ͖

͍ͯɼ͜Ε·Ͱʹଟ͘ͷख๏͕ఏҊ͞Ε͖ͯͨɽ޿͘༻͍ΒΕͳ͍ͨΊɼNMF Ͱ͸Ի‫ݯ‬ͷ௿ϥϯΫੑΛԾఆ͢Δ͜ͱͰԻ‫ݯ‬

͍ͯΔख๏ͷ͏ͪͷҰͭʹɼಠཱ੒෼෼ੳ (ICA) [1] ͕͋ΔɽΛߏ੒͢ΔҼࢠΛਪఆ͠ɼԻ‫ݯ‬෼཭Λߦ͏ɽ۩ମతʹ͸ɼԻ‫ݯ‬

ICA ͸֤Ի‫ݯ‬ͷ౷‫ܭ‬తͳಠཱੑΛԾఆ͢Δ͜ͱʹΑΓࠞ߹ߦ͕‫ج‬ఈͱΞΫςΟϕʔγϣϯͷೋͭͷҼࢠ͔Β੒Δͱ͠ɼ‫؍‬ଌྻͷ‫͋Ͱྻߦٯ‬Δ෼཭ߦྻΛਪఆ͠ɼ෼཭Λߦ͏ɽ͜ͷ ICA

ͷύϫʔεϖΫτϩάϥϜΛ‫ج‬ఈߦྻͱΞΫςΟϕʔγϣϯߦΛ΋ͱʹಠཱϕΫτϧ෼ੳ (IVA) [2] ΍ FastICA [3] ͳͲͷ͞ྻͷੵʹΑΓۙࣅ͢Δɽ͜ͷΑ͏ͳҼࢠ෼ղʹΑΔϞσϧΛ

·͟·ͳख๏͕ఏҊ͞Ε͍ͯΔ͕ɼ͜ΕΒͷख๏͸‫ڞ‬௨ͯ͠ҼࢠϞσϧͱ‫Ϳݺ‬ɽ·ͨɼ͜ͷ NMF ΛϚϧννϟωϧԻ‫ݯ‬෼ϚΠΫ਺͕Ի‫਺ݯ‬ΑΓগͳ͍ྼܾఆ৚݅Ͱ͸෼཭Ͱ͖ͳ͍ͱ

཭ͷͨΊʹ֦ுͨ͠Ϛϧννϟωϧ NMF (MNMF) [12] ΋ఏ͍͏໰୊఺͕͋ΔɽҊ͞Ε͍ͯΔɽMNMF Ͱ͸‫؍‬ଌεϖΫτϩάϥϜΛ‫ج‬ఈߦྻ͜Εʹର͠ɼྼܾఆ৚݅Ͱ΋෼཭͕Մೳͳख๏ͱͯ࣌ؒ͠ɾ

ͱΞΫςΟϕʔγϣϯߦྻͱۭؒಛ௃ྔʹ෼ղ͢ΔɽMNMF

प೾਺ΫϥελϦϯάʹ‫ͮ͘ج‬Ի‫ݯ‬෼཭๏͕ண໨͞Ε͍ͯ͸ۭؒಛ௃ྔ΋Ҽࢠ෼ղʹΑΓਪఆ͢ΔͨΊɼԻ‫ݯ‬ϞσϧɾॏΔ [4–9]ɽ͜ͷΞϓϩʔνͰ͸ɼ֤Ի‫ݯ‬εϖΫτϩάϥϜ͕࣌৞աఔ͕ͱ΋ʹҼࢠϞσϧʹΑΓϞσϧԽ͞Εͨख๏Ͱ͋Δɽ

ؒɾप೾਺ྖҬͰεύʔεͰ͋ΔͱԾఆ͢Δ͜ͱͰɼࠞ߹ԻεຊߘͰ͸ɼԻͷॏ৞աఔͱԻ‫ݯ‬ϞσϧΛ౷߹ͨ͠ϞσϧʹϖΫτϩάϥϜͷ֤࣌ؒɾप೾਺Ϗϯʹ͓͚Δ‫؍‬ଌ͸ͦΕͧΑΓԻ‫ݯ‬෼཭Λߦ͏ɽ·ͨɼैདྷ๏Ͱ͸ॏ৞աఔ͕ࠞ߹ϞσΕ͍ͣΕ͔ҰͭͷԻ‫ݯ‬੒෼͕௚઀‫؍‬ଌ͞Εͨ΋ͷͰ͋ΔͱΈϧͰ΋ҼࢠϞσϧͰ΋ϞσϧԽ͞Ε͍ͯͨ͜ͱ͔Βɼͦͷೋͳ͢ɽ͜ΕʹΑΓɼԻͷॏ৞աఔ͕ࠞ߹ϞσϧʹΑΓද͞ΕͭͷϞσϧԽͷؔ܎ੑʹண໨͠ɼԻ‫ݯ‬Ϟσϧʹ͓͍ͯ΋ࠞ߹Δɽ͜ͷࠞ߹ϞσϧΛਪఆ͢ΔͨΊɼϚΠΫؒͷԻͷҐ૬ࠩϞσϧͱҼࢠϞσϧͷೋͭͷϞσϧԽΛߦͬͨɽͭ·ΓɼҼͱύϫʔࠩΛಛ௃ྔͱͯ͠༻͍ͨࠞ߹ϫτιϯ෼෍ͷΫϥεࢠϞσϧͰ͸Ի‫ݯ‬ύϫʔεϖΫτϩάϥϜͷ֤࣌ؒप೾਺ϏλϦϯά [5–7] ΍֤ϚΠΫͰͷԻͷҐ૬ͱύϫʔΛಛ௃ྔͱ͠

ϯ͕ෳ਺ͷ‫ج‬ఈͱΞΫςΟϕʔγϣϯͷੵͷ࿨Ͱද͞Ε͕ͨɼͯ༻͍ͨࠞ߹Ψ΢ε෼෍ͷΫϥελϦϯά [9, 10] ʹΑΔ෼཭

ࠞ߹ϞσϧͰ͸֤࣌ؒप೾਺ϏϯΛ͍ͣΕ͔Ұͭͷ‫ج‬ఈͱΞ๏͕ఏҊ͞Ε͍ͯΔɽ͜ΕΒͷΫϥελϦϯά͸ͦΕͧΕͷΫςΟϕʔγϣϯͷੵʹΑΓද͢ɽ͜ͷΑ͏ʹɼຊߘͰ͸ॏ࣌ؒप೾਺ϏϯͰͦΕͧΕಠཱʹߦΘΕΔͨΊύʔϛϡςʔ৞աఔɾԻ‫ݯ‬Ϟσϧʹରͯ͠ɼࠞ߹ϞσϧɾҼࢠϞσϧͷͦΕγϣϯ໰୊͕ൃੜ͢Δ͕ɼେ௩Β [9] ͸જࡏతσΟϦΫϨ഑෼ͧΕͷϞσϧԽΛߦ͏͜ͱͰɼͦΕΒͷ૊Έ߹ΘͤʹΑΓෳ๏ (LDA) Λ༻͍֤ͯ࣌ؒप೾਺ϏϯΛ֤Ի‫ׂʹݯ‬Γ౰ͯͨ͋਺ͷԻ‫ݯ‬෼཭๏Λ࣮‫ͨ͠ݱ‬ɽ۩ମతʹ͸ɼਤ 1 ʹࣔ͢Α͏ʹɼͱɼͦΕΒͷԻ‫ݯ‬Λ͞Βʹ͍ͣΕ͔ͷํ޲ʹׂΓ౰ͯΔ͜ͱҼࢠϞσϧͰ͸Ի‫ݯ‬ͷύϫʔεϖΫτϩάϥϜ͕‫ج‬ఈͱΞΫͰ͜ͷύʔϛϡςʔγϣϯ໰୊Λղܾͨ͠ɽ

ςΟϕʔγϣϯͷੵͷ࿨͔Β੒Γɼ‫؍‬ଌεϖΫτϩάϥϜ͸
—1—

ද1

ϞσϧԽํ๏ͷ૊Έ߹Θͤ❳❳❳ॏ৞աఔ ࠞ߹Ϟσϧ ҼࢠϞσϧ❳❳❳Ի‫ݯ‬Ϟσϧ
❳

❳❳❳
❳

ࠞ߹Ϟσϧ

LDA-LDAҼࢠϞσϧ

NMF-LDA NMF-NMF

LDA-NMF!"#$%&'()*+,-

͜ͷͱ͖प೾਺ྖҬͰͷॠ࣌ࠞ߹ΛԾఆ͢ΔͱɼԻ‫ ݯ‬k ͷΈΛ!"#$%&'()*+‫؍‬ଌͨ͠ͱ͖ͷ‫؍‬ଌεϖΫτϧ xtf k ͸ҎԼͷΑ͏ʹද͞ΕΔɽ,-$(a) Ի‫ݯ‬Ϟσϧ

xtf k = af k · ytf kͨͩ͠ɼaf k ͸प೾਺ f ͰͷԻ‫ ݯ‬k ͷ఻ୡؔ਺Ͱ͋Δɽ͜͜!"#$Ͱɼytf k ͕࣍ͷΑ͏ͳෳૉΨ΢ε෼෍ʹै͏ͱ͢Δɽ

%&'

011

()&*'+,-./

!"#$
012

ytf k ∼ NC (0, λtf k )(4)

2
λtf k = E[ytfk ] ͸࣌ࠁ tɼप೾਺ f ͰͷԻ‫ ݯ‬k ͷύϫʔΛද͢ɽ͜ͷͱ͖Ի‫ ݯ‬k ͷΈΛ‫؍‬ଌͨ͠ͱ͖ͷ‫؍‬ଌ xtf k ͸࣍ͷΑ͏ͳ%&'(b) ॏ৞աఔ
ਤ1(3)ෳૉΨ΢ε෼෍ʹै͏ɽఏҊ๏ͷੜ੒ϞσϧɽԻ‫ݯ‬ϞσϧͰ͸֤Ի‫ݯ‬ͷύϫʔεϖΫτ

xtf k ∼ NC (0, λtf k G−1
fk )ϩάϥϜ͕‫ج‬ఈɾΞΫςΟϕʔγϣϯ (ɾϚεΫ) ʹΑΓߏ੒͞
ΕΔɽॏ৞աఔͰ͸ࠞ߹ԻεϖΫτϩάϥϜ͕Ի‫ݯ‬εϖΫτϩ
άϥϜɾ఻ୡؔ਺ (ɾϚεΫ) ʹΑΓߏ੒͞ΕΔɽԻ‫ݯ‬Ϟσϧɾॏ৞աఔͱ΋ʹࠞ߹ϞσϧͰ͸ϚεΫʹΑΓͲΕ͔Ұͭͷ‫ج‬ఈɼ

(5)

͜͜ͰɼG−1
f k ͸प೾਺ f ͰͷԻ‫ ݯ‬k ͷۭؒ૬ؔߦྻͰ͋ΓɼHHG−1͸Τϧϛʔτ‫ڞ‬໾Λࣔ͢ɽ
f k = af k af k Ͱ͋Δɽͨͩ͠ ∗΋͘͠͸Ի‫͕ݯ‬બ୒͞ΕΔ͕ɼҼࢠϞσϧͰ͸ϚεΫ͸ͳ͘͢

2. 2 ॏ ৞ ա ఔ΂ͯͷ‫ج‬ఈɼ΋͘͠͸Ի‫͞༻࢖͕ݯ‬ΕΔɽ

ຊઅͰ͸Իͷॏ৞աఔʹରͯ͠ҼࢠϞσϧͱࠞ߹ϞσϧΛ༻͍ͯϞσϧԽΛߦ͏ɽԻ‫ݯ‬εϖΫτϩάϥϜͱ఻ୡؔ਺͔Βੜ੒͞ΕΔͱ͢Δɽࠞ߹ϞσϧͰ͸Ի‫ݯ‬ͷύϫʔεϖΫτϩάϥϜ͕ϚεΫʹΑͬͯબ୒͞Εͨ‫ج‬ఈͱΞΫςΟϕʔγϣϯ͔Β੒Γɼ‫؍‬ଌεϖ

2. 2. 1ҼࢠϞσϧࠞ߹Իͷ‫؍‬ଌ͕֤Ի‫ͱ͝ݯ‬ͷ‫؍‬ଌͷ࿨ʹ౳͍͠ͱ͢Δͱ
ࣜ (3) ΑΓɼࠞ߹Իͷ‫؍‬ଌ͸࣍ࣜͰ༩͑ΒΕΔɽ

ΫτϩάϥϜ͸ϚεΫʹΑͬͯબ୒͞ΕͨԻ‫ݯ‬εϖΫτϩά
xtf =ϥϜͱ఻ୡؔ਺͔Βੜ੒͞ΕΔͱ͢ΔɽఏҊ๏Ͱ͸ࠞ߹Ϟσϧʹ͸ LDAɼҼࢠϞσϧʹ͸ NMF ͷ࿮૊ΈΛ༻͍ͯΪϒεαϯϓϦϯάʹΑΓԻ‫ݯ‬෼཭Λߦ͏ɽ

2.

K
!

xtf k =

k=1

K
!
k=1

(6)

af k · ytf k͜ͷͱ͖ɼࣜ (5)ɼ(6) ͱΨ΢εͷՃ๏ੑʹΑΓࠞ߹Իͷ‫؍‬ଌ͸࣍ͷΑ͏ͳෳૉΨ΢ε෼෍ʹै͏ͱ͞ΕΔɽϞσϧԽ

xtf | λ, S, G ∼ NCఏҊ๏Ͱ͸ɼԻͷॏ৞աఔͱ֤Ի‫ݯ‬ͷԻ‫ݯ‬ϞσϧΛߟྀ͢Δ
͜ͱͰԻ‫ݯ‬෼཭Λߦ͏ɽϞσϧԽʹ͸ NMF ͷΑ͏ͳҼࢠϞσϧͱ LDA ͷΑ͏ͳࠞ߹Ϟσϧͷೋ௨Γͷํ๏͕͋ΔɽຊষͰ͸ॏ৞աఔͱԻ‫ݯ‬ϞσϧͷͦΕͧΕʹରͯ͜͠ΕΒͷೋ௨ΓͷϞσϧԽΛߦ͏ํ๏ʹ͍ͭͯड़΂ΔɽͦΕΒͷ૊Έ߹ΘͤʹΑΓɼද 1 ʹࣔ͢Α͏ʹԻ‫ݯ‬Ϟσϧ-ॏ৞աఔΛҼࢠ-ࠞ߹Ϟσϧɼࠞ߹-ࠞ߹Ϟσϧɼࠞ߹-ҼࢠϞσϧɼҼࢠ-ҼࢠϞσϧͰͦΕͧΕϞσϧԽͨ͠ NMF-LDAɼLDA-LDAɼLDA-NMFɼ
NMF-NMF ʹ͍ͭͯड़΂Δɽ·ͣɼ୯ҰͷԻ‫ݯ‬Λ‫؍‬ଌͨ࣌͠ͷ‫؍‬ଌϞσϧʹ͍ͭͯड़΂ΔɽఏҊ๏Ͱ͸࣌ؒྖҬͷ৴߸ʹରͯ͠୹࣌ؒϑʔϦΤม‫( ׵‬STFT) Λߦ͏͜ͱʹΑΓಘΒΕΔ࣌ؒप೾਺ྖҬͷ৴߸ʹରͯ͠ϞσϧԽΛߦ͏ɽ·ͣ K ‫ݸ‬ͷԻ‫ݯ‬Λ M ‫ݸ‬ͷϚΠΫΛ༻͍ͯ࿥Ի͢Δͱ͠ɼ࣌ࠁ tɼप೾਺ f Ͱͷࠞ߹Իͷ‫؍‬ଌ
xtf ͱԻ‫ݯ‬৴߸ ytf ΛҎԼͷΑ͏ʹఆٛ͢Δɽ
xtf = [xtf 1 , · · · , xtf M ] ∈ CTytf = [ytf 1 , · · · , ytf K ] ∈ C
T

0,

K
!
k=1

λtf k G−1
fk

#

(7)

͜͜Ͱɼۭؒ૬ؔߦྻ͸Ի‫ݯ‬ͷछྨͰ͸ͳ͘Ի‫ݯ‬ͷํ޲ʹґଘ͢ΔͨΊɼԻ‫ͱ͝ݯ‬ͷۭؒ૬ؔߦྻ Gf k Λํ޲͝ͱͷۭؒ૬
ؔߦྻ Gf d ͷॏΈ෇͖࿨Λ༻͍ͯද͢ɽ͜ͷͱ͖ɼํ޲ d ͷԻ‫ ݯ‬k ΁ͷॏΈΛ rkd ͱ͢Δͱࣜ (7) ͸࣍ͷΑ͏ʹද͞ΕΔɽ
xtf | λ, S, G ∼ NC
2. 2. 2

2. 1 ໰ ୊ ઃ ఆ

"

"

0,

K
D
!
!
k=1 d=1

λtf k rkd G−1
fd

#

(8)

ࠞ߹Ϟσϧ

Ի‫ݯ‬εϖΫτϩάϥϜ͕εύʔεͰ͋Δɼ͢ͳΘͪɼ֤࣌ؒप೾਺Ϗϯʹ͓͍ͯ‫؍‬ଌ͞ΕΔԻ͸ߴʑҰͭͰ͋Δͱ͠ɼͦͷͱ͖ͷࠞ߹Իͷ‫؍‬ଌϞσϧʹ͍ͭͯߟ͑Δɽ·֤ͣ࣌ؒप೾਺Ϗϯʹ͓͍ͯ‫؍‬ଌ͞ΕΔԻ‫ݯ‬Λࣔͨ͢Ίͷม਺Λ
ztf = [ztf 1 , · · · , ztf K ]T ͱ͢Δɽͨͩ͠ɼztf ͸ 1 of K ද‫ݱ‬ͷϕΫτϧͰ͋ΓɼԻ‫ ݯ‬k ͕‫؍‬ଌ͞ΕΔͱ͖͸ ztf k = 1 ͱͳΓɼͦΕҎ֎ͷͱ͖͸ 0 ͱͳΔɽ͜ͷͱ͖ɼ‫؍‬ଌ xtf ͸࣍ࣜͰ༩

M

(1)

K

(2)

͑ΒΕΔɽ

—2—

xtf =

K$z
xtftfkk =

k=1

K
$

k=1

(af k · ytf k )

ztf k

(9)

∼

ࣜ (5)ɼ(9) ΑΓɼ‫؍‬ଌ͸࣍ͷ෼෍ʹ͕ͨͬͯ͠ੜ੒͞ΕΔɽ
xtf ∼

K
$

k=1

2. 4. 2

$
k,d

"

0,

NC

LDA-LDAztf k
NC (0, λtf k G−1
fk )

(10)

∼

ͳ͘ɼํ޲͝ͱʹಠཱͳม਺ Gf d ͱͯ͠ߟ͑Δɽ͜ͷͱ͖ɼࣜ (10) ͸Ի‫ ݯ‬k ͷํ޲Λࣔ͢ϕΫτϧ sk = [sk1 , · · · , skD ]TΛ༻͍Δͱ࣍ͷΑ͏ʹද͞ΕΔɽ

k=1 d=1

2. 4. 3

$

k,d,l

%

NC 0, wklf hklt G−1
fd

NMF-NMF&ztf k skd utf kl(15)

NMF-NMF ͸ॏ৞աఔ͕ҼࢠϞσϧɼԻ‫ݯ‬Ϟσϧ͕ҼࢠϞσϧͳͷͰࣜ (8)ɼ(12) ΑΓ໬౓ؔ਺͸࣍ͷΑ͏ʹͳΔɽ

ztf k skd
NC (0, λtf k G−1
fd )

(11)

xtf | W ,H, R, G

ͨͩ͠ɼsk ͸ 1 of D ද‫ݱ‬ͷϕΫτϧͰ͋ΓɼԻ‫ ݯ‬k ͕ํ޲ d
2. 3 Ի‫ݯ‬Ϟσϧ

2. 4. 4ຊઅͰ͸Ի‫ ݯ‬k ͷύϫʔ λtf k ʹରͯ͠ҼࢠϞσϧͱࠞ߹ϞσϧΛ༻͍ͯϞσϧԽΛߦ͏ɽ

"

0,

∼ NC

ʹ͋Δͱ͖͸ skd = 1ɼͦΕҎ֎ͷͱ͖͸ 0 ͱͳΔɽLDA-NMF!

wklf hklt rkd G−1
fd

k,d,l

#

(16)

LDA-NMF ͸ॏ৞աఔ͕ࠞ߹ϞσϧɼԻ‫ݯ‬Ϟσϧ͕ҼࢠϞσϧͳͷͰࣜ (11)ɼ(12) ΑΓ໬౓ؔ਺͸࣍ͷΑ͏ʹͳΔɽҼࢠϞσϧԻ‫ ݯ‬k ͷύϫʔεϖΫτϩάϥϜ λtf k ͕௿ϥϯΫੑΛ࣋ͭͱԾఆ͢ΔͱɼύϫʔεϖΫτϩάϥϜ͸‫ج‬ఈεϖΫτϧ wklf
ͱΞΫςΟϕʔγϣϯ hklt ͷੵͷ࿨Ͱද‫͞ݱ‬ΕΔɽ͜ͷ‫ج‬ఈεϖΫτϧͱΞΫςΟϕʔγϣϯ͸ͦΕͧΕ֤‫ج‬ఈͷप೾਺తಛ௃ͱͦͷ‫ج‬ఈͷ֤࣌ࠁ͝ͱͷԻͷେ͖͞Λࣔ͢ɽ‫ج‬ఈͷ਺Λ L ͱ͢ΔͱύϫʔεϖΫτϩάϥϜ͸࣍ࣜͰද͞ΕΔɽ
λtf k =!

wklf hklt(12)

xtf | W ,H, R, G
∼

$
l

NC

"

0,

!

wklf hklt rkd G−1
fdk,d#utf kl(17)͔͠͠ɼ͜ͷϞσϧͰ͸Ի‫ ݯ‬k ʹؔ͢Δύϥϝʔλ͸Ψ΢ε
෼෍ͷ෼ࢄύϥϝʔλͱͯ͠Ҽࢠ෼ղʹΑΓਪఆ͞ΕΔ΂͖
Ͱ͋Δ͕ɼ‫ج‬ఈͷબ୒Λߦ͏ utf kl ͕ࠞ߹Ψ΢ε෼෍ʹ͓͚Δ
෼෍Λબ୒͢Δύϥϝʔλͱͯ͠ଘࡏ͍ͯ͠Δɽ͜ΕʹΑΓɼl2. 3. 2(14)ϧͳͷͰࣜ (11)ɼ(13) ΑΓ໬౓ؔ਺͸࣍ͷΑ͏ʹͳΔɽ

ଘ͢ΔͨΊɼۭؒ૬ؔߦྻ Gf k ΛԻ‫ʹͱ͝ݯ‬ಠཱͳม਺Ͱ͸

2. 3. 1#ztf k skd

LDA-LDA ͸ॏ৞աఔ͕ࠞ߹ϞσϧɼԻ‫ݯ‬Ϟσϧ͕ࠞ߹Ϟσxtf | W ,H, Z, S, U , G

K D
$
$

wklf hklt G−1
fdl͜͜Ͱɼۭؒ૬ؔߦྻ͸Ի‫ݯ‬ͷछྨͰ͸ͳ͘Ի‫ݯ‬ͷํ޲ʹґ

xtf ∼

!

utf kl ͸Ҽࢠ෼ղͰ͸ਪఆ͢Δ͜ͱ͕Ͱ͖ͳ͍ͨΊ͜ͷϞσϧԽํ๏Ͱ͸ LDA-NMF Λ࣮‫͢ݱ‬Δ͜ͱ͕Ͱ͖ͳ͍ɽ

ࠞ߹Ϟσϧࠞ߹ϞσϧͰ΋ҼࢠϞσϧͷΑ͏ʹ‫ج‬ఈεϖΫτϧ wklf ͱ

2. 5 ࣄલ෼෍ͷઃ‫ܭ‬ΞΫςΟϕʔγϣϯ hklt Λ༻͍ͯύϫʔεϖΫτϩάϥϜ

ఏҊ๏Ͱ͸ࣜ (14)–(17) ͷύϥϝʔλʹର͠ɼͧΕͧΕద੾

λtf k Λද‫͢ݱ‬Δɽͨͩ͠ɼࠞ߹ϞσϧͰ͸શͯͷ‫ج‬ఈͷ࿨Ͱͳࣄલ෼෍Λ༩͑Δ͜ͱͰਪ࿦Λߦ͏ɽ·ͣɼztf ɼsk ɼutf kύϫʔΛද‫͢ݱ‬Δͷʹର͠ɼఏҊ๏Ͱ͸֤࣌ؒप೾਺Ϗϯ͝͸ΫϥελϦϯάʹΑΔਪ࿦Λߦ͏ͨΊʹΧςΰϦΧϧ෼෍′ͱʹҰͭͷ‫ج‬ఈ l Λબ୒͢Δ͜ͱʹΑΓύϫʔ λtf k Λ
λtf k = wkl′ f hkl′ t͔Βੜ੒͞ΕΔͱ͢Δɽ
(13)ͱ ද ͢ ɽͦ ͷ ‫ ج‬ఈ Λ બ ୒ ͢ Δ ͨ Ί ͷ ม ਺ Λ utf k=[utf k1 , · · · , utf kL ]T ͱ͢Δɽͨͩ͠ɼutf k ͸ 1 of L ද‫ݱ‬ͷϕΫτϧͰ͋Γɼ‫ج‬ఈ l ͕༻͍ΒΕΔͱ͖͸ utf kl = 1ɼͦΕҎ֎ͷͱ͖͸ 0 ͱͳΔɽ
2. 4 ֤Ϟσϧͷ໬౓ؔ਺ຊষͰ͸ɼԻ‫ݯ‬Ϟσϧ-ॏ৞աఔΛҼࢠ-ࠞ߹Ϟσϧɼࠞ߹ࠞ߹ϞσϧɼҼࢠ-ҼࢠϞσϧɼࠞ߹-ҼࢠϞσϧͰͦΕͧΕϞσϧԽͨ͠ NMF-LDAɼLDA-LDAɼNMF-NMFɼLDA-NMF
ͷ‫؍‬ଌϞσϧʹ͍ͭͯड़΂Δɽ
2. 4. 1NMF-LDANMF-LDA ͸ॏ৞աఔ͕ࠞ߹ϞσϧɼԻ‫ݯ‬Ϟσϧ͕ҼࢠϞσϧͳͷͰࣜ (8)ɼ(13) ΑΓ໬౓ؔ਺͸࣍ͷΑ͏ʹͳΔɽ

ztf | πt ∼ Categorical(πt )(18)

utf k | ψtk ∼ Categorical(ψtk )(20)

sk | φ ∼ Categorical(φ)(19)͜͜ͰɼϋΠύʔύϥϝʔλ πt ɼφɼψtk ͸‫؍‬ଌʹґଘͯ͠ม
ಈ͢ΔͨΊਪ࿦Λඞཁͱ͢Δɽ͕ͨͬͯ͠ πt ɼφɼψtk ͸ΧςΰϦΧϧ෼෍ͱ‫ڞ‬໾ͳσΟϦΫϨ෼෍͔Βੜ੒͞ΕΔͱ͢Δɽ
πt ∼ Dirichlet(aπ0 1K )(21)

ψtk ∼ Dirichlet(aψ
0 1L )(23)φ ∼ Dirichlet(aφ0 1D )(22)

͜͜Ͱɼ1N ͸ཁૉ͕શͯ 1 ͷ N ࣍‫ݩ‬ͷϕΫτϧͱ͠ɼa∗0 ͸ϋΠύʔύϥϝʔλͱ͢Δɽ·ͨɼۭؒ૬ؔߦྻ Gf d ɼ‫ج‬ఈ
wklf ɼΞΫςΟϕʔγϣϯ hklt ɼॏΈ rkd ͸ࣜ (14)–(17) ͱ‫ڞ‬

xtf | W ,H, Z, S, G
—3—໾ʹͳΔΑ͏ʹࣄલ෼෍ͱͯ࣍͠ͷΑ͏ͳ෼෍Λ༩͑Δɽ
Gf d ∼ WC (ν, G0f d )(24)hklt ∼ Gamma(ah0 , bh0 )(26)

w
wklf ∼ Gamma(aw0 , b0 )′ͨɼϋΠύʔύϥϝʔλ νf′ d ɼG′f d ɼπtf
ɼφ′k ͸࣍ͷΑ͏ʹͳΔɽ

νf′ d = ν +

′
πtf
k =

֤Ի‫ݯ‬৴߸͸ɼॏ৞աఔ͕ҼࢠϞσϧͷͱ͖͸Ϛϧννϟ
ωϧ Wiener ϑΟϧλɼࠞ߹Ϟσϧͷͱ͖͸࣌ؒप೾਺ϚεΩϯάʹΑΓ෮‫͢ݩ‬Δ͜ͱ͕Ͱ͖ΔɽϚϧννϟωϧ WienerϑΟϧλͰ͸Ի‫ ݯ‬k ͷ৴߸ xktf ͸ҎԼͷࣜʹΑΓ෮‫͞ݩ‬ΕΔɽ
xktf = Ytf k Ytf−1 xtf
d,lwklf hklt rkd G−1
f d ɼYtf =

(28)

'

k

Ytf k Ͱ͋Δɽ࣌ؒप೾਺ϚεΩϯάͰ͸ɼ࣌ؒप೾਺ྖҬͰԻ‫͝޲ํݯ‬ͱʹϚεΫΛਪఆ͢Δ͜ͱʹΑΓ෼཭ԻΛੜ੒͢ΔɽΪϒεαϯϓϦϯάͰ i ճ໨ͷࢼߦʹΑΓಘΒΕΔαϯϓϧΛ
W

(i)

,H

(i)

(i)

,G ,Z

(i)

,S

(i)

, (U

(i)

) ͱ͢Δͱɼ࣌ࠁ t प೾਺ f ͰdͷԻ‫ ޲ํݯ‬d ʹର͢ΔϚεΫ Mtf
͸࣍ͷΑ͏ʹͳΔɽ

1
d
Mtf
=
I

I
K
!
!
(i)

(i)
ztf k skd

(29)

l

d

l

(36)

)
*
+,ztf k$ ()))
xH
tf Gf d xtf) ' Gf d) exp − 'φ′kd =)
wklf hklt )wklf hklt
l

t,f

l

× (aφ0 + c¬k
d )

(37)

͜͜Ͱɼn¬tf
tk ͸࣌ࠁ t प೾਺ f ͰͷαϯϓϧΛআ͍ͯɼ࣌ࠁ tʹ͓͍ͯԻ‫ ݯ‬k ʹׂΓ౰ͯΒΕͨ࣌ؒप೾਺Ϗϯͷ਺Λද͠ɼc¬kd ͸Ի‫ ݯ‬k Λআ͍ͯํ޲ d ʹׂΓ౰ͯΒΕͨԻ‫ݯ‬ͷ਺Λද͢ɽ·ͨɼntkl ͸࣌ࠁ t ʹ͓͍ͯԻ‫ ݯ‬kɼ‫ج‬ఈ l ʹׂΓ౰ͯΒkΕͨ࣌ؒप೾਺Ϗϯͷ਺Λࣔ͠ɼn¬tf
͸ ntkl ͔Β࣌ࠁ t प೾tkl

਺ f ͰԻ‫ ݯ‬k ʹׂΓ౰ͯΒΕͨཁૉΛআ͍ͨ΋ͷͰ͋Δɽ
wklf ͱ hklt ͸ࣄ‫ޙ‬෼෍͔Β௚઀αϯϓϧΛಘΔ͜ͱ͸೉͍ͨ͠Ίɼิॿؔ਺๏Λ༻͍ͯಘΒΕΔԼ‫͔ݶ‬ΒαϯϓϦϯά
͖ɼ΋ͱͷର਺໬౓ͱҰக͢Δɽ·ͣɼର਺໬౓͸࣍ࣜͰ༩͑
ΒΕΔɽ

i=1 k=1dxdtf = Mtf
xtf

c

(30)

ʹΑΓ෮‫͞ݩ‬ΕΔɽ

ਪ

)
*
+,skd
$ ())
)
xH
tf Gf d xtf) ' Gf d) exp − ')wklf hklt )wklf hklt

Λߦ͏ɽͨͩ͠ɼ͜ͷԼ‫ݶ‬͸ิॿม਺ʹ͍ͭͯ࠷େԽͨ͠ͱ

͜ͷϚεΫΛ༻͍Δͱํ޲ d ͷԻ‫ݯ‬ͷ৴߸ xdtf ͸

3.

(35)

wklf hklt

× (aπ0 + n¬tf
tk )2. 6 ෼཭Իͷੜ੒

ͨͩ͠ɼYtf k =lt,k΢Ογϟʔτ෼෍ (෇࿥ࢀর) ͱ͢Δɽ'! xtf xtf H'
ztf k skd

0 −1
G′−1
+
f d = (Gf d )(27)͜͜Ͱɼνɼa∗0 ɼb∗0 ͸ϋΠύʔύϥϝʔλͰ͋ΓɼWC ͸ෳૉ

(34)

ztf k skd

t,k

(25)

rkd ∼ Gamma(ar0 , br0 )!

log p(X | W , H, G, Z, S) =!

ztf k skd

t,f,k,d))#)!
) xH G x)) 'tf f d tf
−M log )wklf hklt ) −
w h
) l
)
l klf klt

"

(38)

࿦

ఏҊ๏Ͱ͸ɼ‫؍‬ଌσʔλू߹ X ʹର͢Δ͢΂ͯͷύϥϝʔλͷࣄ‫ޙ‬෼෍ p(G, W , H, (Z, S, R, U , π, φ, ψ)|X) Λ࠷େͱ͢ΔύϥϝʔλΛ‫ٻ‬ΊΔ͜ͱΛ໨ඪͱ͢Δɽͨͩ͠ɼ͜ΕΒͷ͜ͷର਺໬౓ʹର͠ςΠϥʔల։ͱ Jensen ͷෆ౳ࣜΛ༻͍ͯԼ‫ݶ‬Λ‫͢ࢉܭ‬Δ [13]ɽ·ͣҰ࣍ͷςΠϥʔల։ʹΑΓɼತؔ਺ f (y) = − log y ͷԼ‫ݶ‬͸ҎԼͷΑ͏ʹ‫͞ࢉܭ‬ΕΔɽ

ύϥϝʔλΛղੳతʹ‫ٻ‬ΊΔ͜ͱ͸ࠔ೉ͳͷͰɼఏҊ๏Ͱ͸y− log y >= − log α + 1 − α͜ΕΒͷύϥϝʔλΛΪϒεαϯϓϦϯάʹΑΓ‫ٻ‬ΊΔ͜ͱα ͸ิॿม਺Ͱ͋Γɼ౳߸͸ α = y ͷͱ͖੒ཱ͢Δɽ࣍ʹͱ͢Δɽͨͩ͠ɼఏҊ๏Ͱ͸ πɼφɼψ ͸ੵ෼ফ‫ڈ‬Λߦ͍ɼ࢒Jensen ͷෆ౳ࣜʹΑΓɼԜؔ਺ g(y) = − y1 ʹର࣍͠ͷෆ౳ࣜΓͷύϥϝʔλ Θ = {G, W , H, (Z, S, R, U )} Λ‫ٻ‬ΊΔ͜ͱ͕੒ཱ͢Δɽ

ͱ͢ΔɽΪϒεαϯϓϦϯάͰ͸ͦΕͧΕͷύϥϝʔλͷࣄβ k ͸ βk >= 0 ͔ͭ

3. 1 NMF-LDA ͷߋ৽
NMF-LDA Ͱ͸ Gf d ɼztf , sk ͸໬౓ͱࣄલ෼෍ͷੵʹΑΓಘΒΕΔҎԼͷࣄ‫ޙ‬෼෍͔Βαϯϓϧ͢Δ͜ͱʹΑΓߋ৽͞
ΕΔɽ

−'

1

>−
y =
k k

‫ޙ‬෼෍Λ‫ٻ‬ΊɼͦΕΒͷࣄ‫ޙ‬෼෍͔ΒαϯϓϦϯάΛ‫܁‬Γฦ͢͜ͱʹΑΓਪఆΛߦ͏ɽ

(39)

'

͸ β k = yk (

k

'

k
−1

yk )

!

βk2

k

1
yk

(40)

βk = 1 Λຬͨ͢ิॿม਺Ͱ͋Γɼ౳߸ͷͱ͖੒ཱ͢Δɽ͜ΕΒͷೋͭͷෆ౳ࣜʹΑΓର਺໬౓ͷԼ‫ݶ‬͸࣍ࣜʹΑΓ‫ٻ‬ΊΒΕΔɽc

Gf d | X, Θ¬Gf d ∼ WC (νf′ d , G′f d )′ztf | X, Θ¬ztf ∼ Categorical(πtf)

sk | X, Θ¬sk ∼ Categorical(φ′k )(31)

log p(X | W , H,G, Z, S) >=(32)

ztf k skd

(33)

͜͜ͰɼΘ¬∗ ͸ Θ ͔Β ∗ ͷཁૉͷΈΛআ͍ͨू߹ͱ͢Δɽ·!

t,f,k,d*−M*

log |αtf | − 1 +−xH
tf Gf d xtf

!
l

'

βtf l
wklf hkltlwklf hklt
αtf

#

+

(41)

—4—

͜͜Ͱɼαtf ͱ βtf l ͸ิॿม਺Ͱ͋Γɼ࣍ͷࣜΛຬͨ͢ͱ͖౳

φ′kd =

߸͕੒ཱ͢Δɽ
αtf =

!

t,f,l

(42)

wklf hkltl

βtf l = wklf hklt"!

wklf hklt

l#−1

× (aφ0 + c¬k
d )

(43)

d

¬tf k
× (aψ
0 + ntkl )hhhklt | X, Θ¬hklt ∼ GIG(γklt, ρhklt , τklt
)

(44)

w
τklf
=

(45)

t,d

!
t,d

αtf k

2
βtf
kl
tr(Xtf Gf d )
hklt

h
γklt
= ah0

! M wklf
f,d

h
τklt
=

!
f,d

αtf k

2
βtf
kl
tr(Xtf Gf d )
wklf

h
τklt
=

(47)

(68)

ztf k skd utf kl

wklf

͜͜Ͱɼnf kl ͸प೾਺ f ʹ͓͍ͯԻ‫ ݯ‬kɼ‫ج‬ఈ l ʹׂΓ౰ͯΒ
(48)

(50)

Εͨ࣌ؒप೾਺Ϗϯͷ਺Λࣔ͢ɽ
3. 3 NMF-NMF ͷߋ৽
NMF-NMF Ͱ͸Ͳͷύϥϝʔλ΋ࣄ‫ޙ‬෼෍͔Β௚઀αϯϓϧΛಘΔ͜ͱ͸೉͍ͨ͠Ίɼิॿؔ਺๏Λ༻͍ͯಘΒΕΔԼ
‫͔ݶ‬ΒαϯϓϦϯάΛߦ͏ɽͨͩ͠ɼ͜ͷิॿؔ਺͸ิॿม਺ʹ͍ͭͯ࠷େԽͨ͠ͱ͖ɼ΋ͱͷର਺໬౓ͱҰக͢Δɽ·(51)ͣɼର਺໬౓͸࣍ࣜͰ༩͑ΒΕΔɽ
clog p(X|W , H, R,G) =!%LDA-LDA Ͱ͸͢΂ͯͷύϥϝʔλ͕ɼ໬౓ؔ਺ͱࣄલ෼෍
ͷੵʹΑΓಘΒΕΔࣄ‫ޙ‬෼෍͔Βαϯϓϧ͢Δ͜ͱ͕ՄೳͰ
͋Δɽࣄ‫ޙ‬෼෍͸࣍ͷΑ͏ʹఆΊΒΕΔɽ′ztf | X, Θ¬ztf ∼ Categorical(πtf)(67)! xtf Gf d xtf
f,d3. 2 LDA-LDA ͷߋ৽

Gf d | X, Θ¬Gf d ∼ WC (νf′ d , G′f d )(66)

ρhklt = bh0(49)

ρhklt = bh0 +(65)

ztf k skd utf kl

hklt

h
γklt
= ah0 − M ntkl

(46)

w
ρw
klf = b0 +(64)

! xtf Gf d xtft,dϋΠύʔύϥϝʔλ γ∗∗ ɼρ∗∗ ɼτ∗∗ ͸࣍ͷΑ͏ʹͳΔɽ

! M hklt

(63)

w
ρw
klf = b0

͜͜ͰɼGIG ͸ҰൠԽ‫ٯ‬Ψ΢ε෼෍ (෇࿥ࢀর) Λࣔ͢ɽ·ͨɼ
w
γklf
= aw
0

(62)

w
γklf
= aw
0 − M nf kl

෼෍͔Βαϯϓϧ͞ΕΔɽ
w
w
wkf l | X, Θ¬wklf ∼ GIG(γklf, ρw
kf l , τkf l )

(61)

)
* H
+,ztf k skd$ ()) Gf d )xtf Gf d xtf
′
)
)
ψtf
=
exp
−
kl
) wklf hklt )wklf hklt

wklf ͱ hklt ͸ࣜ (25)ɼ(26)ɼ(41) ͷੵʹΑΓಘΒΕΔ࣍ͷ

w
τklf
=

* H
+,ztf k utf kl$ ()) Gf d )))) exp − xtf Gf d xtf) wklf hklt )wklf hklt

(52)
(53)

tf

&

− log |Ytf | − tr(Xtf Ytf−1 )

(69)

−1
͜͜Ͱɼ Xtf = xH
tf xtf ɼYtf kld = wklf hklt rkd Gf d ɼYtf ='kld

Ytf kld ͱͨ͠ɽࣜ (69) ͷԼ‫ݶ‬Λ‫ٻ‬ΊΔͨΊʹೋͭͷෆ౳ࣜΛ༻͍Δ [14]ɽ·ͣɼଟมྔͷತؔ਺ f (Y ) = − log |Y |(Y ≽ 0 ∈ CM ×M ) ʹରͯ͠Ұ࣍ͷଟมྔͷςΠϥʔల։Λ༻
͍Δͱ࣍ͷෆ౳͕ࣜಘΒΕΔɽ

sk | X, Θ¬sk ∼ Categorical(φ′k )

(54)

w
w
wklf | X, Θ¬wklf ∼ GIG(γklf, ρw
klf , τklf )(56)

౳ ߸ ͸ Ω = Z ͷ ͱ ͖ ੒ ཱ ͢ Δ ɽ࣍ ʹ ɼԜ ؔ ਺ g(Y ) =(57)−tr(Y −1 A) ʹରͯ࣍͠ͷෆ౳͕ࣜ੒Γཱͭɽ′
utf kl | X, Θ¬utf kl ∼ Categorical(ψtfk)

hhhklt | X, Θ¬hklt ∼ GIG(γklt, ρhklt , τklt)′͜͜ͰɼϋΠύʔύϥϝʔλ νf′ d ɼG′f d ɼπtf
ɼφ′k ɼψɼγ ∗ ɼρ∗ ɼ∗τ ͸࣍ͷΑ͏ʹͳΔɽ
νf′ d = ν +

!

(58)

ztf k skd

t,k

0 −1
G′−1
+
f d = (Gf d )! xtf xtf H
t,k,l

′
πtf
k =

wklf hklt

−tr

"" K
!

Yk

k=1

#−1 #
A

(59)

d,l

(60)

>
=−

K
!

'

k=1

k′

Zk′ )

−1

%

(70)

tr Yk−1 Φk AΦHk'͜͜Ͱ {Φk }K
k=1 ͸ิॿม਺Ͱ͋Γɼͨɼ౳߸͸ Φk = Zk (ztf k skd utf kl* H+,skd utf kl$ ()) Gf d )))) exp − xtf Gf d xtf) wklf hklt )wklf hklt

× (aπ0 + n¬tf
tk )−1
− log |Y | >= − log |Ω| − tr(Ω Y ) + M

(55)

k

&

(71)

Φk = I Λຬͨ͢ɽ·ͷͱ͖੒ཱ͢Δɽෆ౳ࣜ (70)ɼ(71) Λ༻͍Δͱର਺໬౓ (ࣜ (69)) ͷԼ‫ݶ‬͸࣍ͷΑ͏ʹ‫ٻ‬ΊΒΕΔɽlog p(X|W , H, S, G)
c

>
=

!%
tf

−

%

&

−tr Ytf Ω−1− log |Ωtf | + M
tf!

tf kld%tr Ytf−1
kld Φtf kld Xtf Φtf kld

&

&

(72)

—5—

͜͜Ͱɼิॿม਺ Ωtf ͱ Φtf kld ͕࣍ࣜΛຬͨ͢ͱ͖౳߸͕੒

0

ཱ͢Δɽ
Ωtf = Ytf
Φtf kld = Ytf kld Ytf−1

(73)

1

(74)

100
80
2

80
wklf ɼhklt ɼskd ɼGf d ͸ࣜ (24)–(27)ɼ(72) ͷੵʹΑΓಘΒ

428

ΕΔ࣍ͷ෼෍͔Βαϯϓϧ͞ΕΔɽ
w
w
wklf | X, Θ¬wklf ∼ GIG(γklf, ρw
klf , τklf )

(75)

r
r
rkd | X, Θ¬rkd ∼ GIG(γkd, ρrkd , τkd
)

(77)

h
h
hklt | X, Θ¬hklt ∼ GIG(γklt, ρhklt , τklt)

(76)

Gf d | X, Θ¬Gf d ∼ MGIGC (ν0 , Rf d , Uf d )(78)

͜͜ͰɼMGIGC ͸ complex matrix generalized inverse Gaussian (෇࿥ࢀর) Λࣔ͢ɽ͜͜ͰɼϋΠύʔύϥϝʔλ γ∗∗ ɼρ∗∗ ɼτ∗∗ ɼRf d ɼUf d ͸࣍ͷΑ͏ʹఆΊΒΕΔɽ
w
γklf
= aw
0

(79)

w
ρw
klf = b0 +!

−1hklt rkd tr(G−1f d Ωtf )

(80)

td

w
τklf
=

! −1 −1hklt rkd tr (Gf d Φtf kld Xtf Φtf kld )

(81)

td

h
γklt
= ah0

ρhklt = bh0 +
h
τklt
=

!

!

(82)

−1
wklf rkd tr(G−1f d Ωtf )

(83)

fd

−1 −1
wklf
rkd tr (Gf d Φtf kld Xtf Φtf kld )

(84)

fd

ρrkd = br0 +

!

!

−1
wklf hklt tr(G−1f d Ωtf )

−1 −1
wklf
hklt tr (Gf d Φtf kld Xtf Φtf kld )Rf d = (G0f d )−1 +Uf d =εจ [17] ͔Βબ୒ͨ͠ɽαϯϓϦϯάप೾਺͸ 16 kHz ͱ͠ɼ
STFT Ͱ͸૭෯ 512 ͷϋϛϯά૭Λγϑτ෯ 256 Ͱ࢖༻ͨ͠ɽ
‫ج‬ఈ਺ L = 20 ͱ͠ɼϋΠύʔύϥϝʔλ͸ɼν = M + 1ɼ
w
h
r
w
h
r
aπ0 = aφ0 = 10ɼaψ
0 = a0 = a0 = a0 = b0 = b0 = b0 = 1 ͱ͠
−1ͨɽ·ͨɼG0f d = (af d aH
ͱ͠ɼaf d ʹ͸ແ‫ڹ‬
f d + 0.01 × I)ࣨͰ 5◦ ִؒͰ࿥Իͨ͠ΠϯύϧεԠ౴Λ༻͍ͨɽΪϒεαϯ
ϓϦϯάͷࢼߦճ਺͸ 200 ճͱ͠ɼ͸͡Ίͷ 180 ճ͸ burn-inͱͯ͠‫غ‬٫ͨ͠ɽධՁई౓ͱͯ͠ɼSignal-to-distortion ratio (SDR)ɼSignal-to-inference ratio (SIR)ɼSignal-to-artificial
ratio (SAR) Λ༻͍ͨɽSDR ͸૯߹తͳ෼཭ੑೳɼSIR ͸໨తԻҎ֎ͷԻͷআ‫ڈ‬ੑೳɼSAR ͸෼཭Իͷ࿪Έͷগͳ͞Λද͢ई౓Ͱ͋Δɽ
4. 2 ࣮ ‫ ݁ ݧ‬Ռද 2ɼ3ɼ4 ʹ࣮‫݁ݧ‬ՌΛࣔ͢ɽͦΕͧΕͷ৚݅ʹ͓͍ͯ࠷΋਺஋͕େ͖͘ͳͬͨ΋ͷΛଠࣈͰࣔͨ͠ɽSIR ͸ LDA-LDA͕࠷΋ߴ͍ੑೳΛ͕ࣔͨ͠ɼSDR ͱ SAR ͸ NMF-NMF ͕࠷ͱͯ͠࢖༻͢ΔϚΠΫϩϗϯΞϨΠͷΠϯύϧεԠ౴Λඞཁ

(86)

ͱ͢Δ͕ɼແ‫Ͱࣨڹ‬࿥Իͨ͠ΠϯύϧεԠ౴Λ༻͍ͯ΋࢒‫ڹ‬(87)஌ͷ‫ڥ؀‬Լʹ͓͍ͯ΋࢖༻͢ΔϚΠΫϩϗϯΞϨΠ͕ಉ͡Ͱ!

࣌ؒ 400ms ͷ‫ڥ؀‬ԼͰͷࠞ߹Իͷ෼཭͕Ͱ͖ͨ͜ͱ͔Βɼະ͋Ε͹ఏҊ๏ʹΑΓԻ‫ݯ‬෼཭͕ՄೳͰ͋Δͱߟ͑ΒΕΔɽ

−1 −1 −1wklf
hklt rkd Φtf kld Xtf Φtf kld

(88)

tkl

wklf hklt rkd Ω−1
tf

(89)

tkl

4.

ϚΠΫ഑ஔ

(85)

tf l

tf l

!

ਤ2

΋ߴ͍ੑೳΛࣔͨ͠ɽ·ͨɼఏҊ๏Ͱ͸ࣄલ෼෍ͷύϥϝʔλ

r
γklf
= ar0

r
τkd
=

3

ධՁ࣮‫ݧ‬

ఏҊ๏ͷ෼཭ੑೳΛධՁ͢ΔͨΊɼγϛϡϨʔγϣϯʹΑΓ
ࠞ߹ͨ͠ԻΛ༻͍࣮ͨ‫ݧ‬Λߦͬͨɽൺֱख๏ͱͯ͠ɼIVA [15]ɼ
Ϛϧννϟωϧ NMF (MNMF) [12]ɼԻ‫ݯ‬ϞσϧΛεύʔεͱۭؒ͠Ϟσϧʹ LDA Λ༻͍ͨ෼཭๏ (LDA) [9] Λ༻͍ͨɽ·ͨɼLDA Ͱ͸Ի‫਺ݯ‬ͷಉ࣌ਪఆ΋ߦ͏͕ɼ৚݅Λର౳ʹ͢
ΔͨΊԻ‫਺ݯ‬͸‫ط‬஌ͱͨ͠ɽ
4. 1 ࣮ ‫ ݧ‬৚ ݅

5.

͓ΘΓʹ

ຊߘͰ͸ɼԻ‫ݯ‬Ϟσϧɾॏ৞աఔͷͦΕͧΕʹରͯ͠ɼҼࢠϞσϧͱࠞ߹ϞσϧΛ༻͍ͨϞσϧԽΛߦ͏͜ͱͰෳ਺ͷԻ‫ݯ‬෼཭๏ΛఏҊͨ͠ɽͦΕͧΕͷϞσϧԽʹΑΔԻ‫ݯ‬෼཭๏Λൺֱ͢ΔͱɼSDRɼSAR ͷ‫Ͱ఺؍‬͸Ի‫ݯ‬Ϟσϧͱॏ৞աఔ
͕‫ʹڞ‬ҼࢠϞσϧʹΑΓϞσϧԽ͞Εͨख๏͕࠷΋ੑೳ͕ߴ
͘ɼSIR ͷ‫Ͱ఺؍‬͸Ի‫ݯ‬Ϟσϧͱॏ৞աఔ͕‫߹ࠞʹڞ‬ϞσϧͰϞσϧԽ͞Εͨख๏͕࠷΋ੑೳ͕ߴ͍͜ͱΛ֬ೝͨ͠ɽͨ
ͩ͠ɼԻ‫ݯ‬Ϟσϧ͕ࠞ߹Ϟσϧɼॏ৞աఔ͕ҼࢠϞσϧͰϞσϧԽ͞Εͨख๏͸·࣮ͩ‫͓͖ͯͰݱ‬Βͣɼࠓ‫ޙ‬͸ͦͷϞσϧԽΛߦ͏ํ๏ʹ͍ͭͯߟ͑Δඞཁ͕͋Δɽ

ਤ 2 ʹԻ‫ͱݯ‬ϚΠΫͷ഑ஔΛࣔ͢ɽ࢒‫ ؒ࣌ڹ‬400 ms ͷΠϯύϧεԠ౴Λ༻͍ͯ 3 Ի‫ݯ‬Λࠞ߹ͨ͠Ի੠Λ༻͍ͨɽϚΠΫ਺͸ 4 ͱͨ͠ɽࠞ߹Իʹ͸ɼԻ੠ͷΈͷࠞ߹ԻͱԻָͷΈͷࠞ߹ԻɼԻ੠ͱԻָͷࠞ߹ԻΛͦΕͧΕ 10 ‫͠༻࢖ͭͣݸ‬ͨɽ༻͍ΔԻָͱԻ੠͸ SISEC [16] ͱ JNAS ͷԻૉόϥϯ෇࿥ෳૉ΢Ογϟʔτ෼෍ͱҰൠԽ‫ٯ‬Ψ΢ε෼෍ͷ֬཰ີ౓ؔ
਺͸ҎԼͷͱ͓ΓͰ͋Δɽ
WC (G|ν, G0 ) =|G|ν−M exp(−tr(G(G0 )−1 ))|G0 |ν π M (M −1)/2

-M −1
m=0

Γ(ν − m)

(A·1)

—6—

ද2

[9]

ԻָʹΑΔධՁ

SDR

SIR

SAR

IVA

0.3 dB

4.9 dB

5.7 dB

MNMF

1.0 dB

6.2 dB

6.7 dB[10]LDA

0.7 dB

7.4 dB

4.1 dB

NMF-LDA

0.5 dB

8.7 dB

3.2 dB

LDA-LDA

0.7 dB

8.7 dB

3.3 dB

NMF-NMF

3.2 dB

8.1 dB

7.5 dBද3Ի੠ʹΑΔධՁ

SDR

SIR

[11]

[12]
SAR

IVA

3.4 dB

7.5 dB

7.1 dB

MNMF

4.8 dB

10.0 dB

7.7 dB

LDA

5.5 dB

15.1 dB

6.3 dB

NMF-LDA4.2 dB

14.0 dB

5.2 dB

LDA-LDA

5.8 dB

17.0 dB

6.3 dB

NMF-NMF

6.0 dB

12.6 dB

7.5 dB

ද4

[13]

[14]

Իָ + Ի੠ʹΑΔධՁ

SDR

SIR

SAR

[15]

IVA

0.1 dB

5.3 dB

5.3 dB

MNMF

1.8 dB

8.6 dB

6.1 dB

LDA

2.4 dB

11.5 dB

4.5 dB

NMF-LDA

1.1 dB

9.8 dB

4.1 dB

LDA-LDA2.8 dB

14.2 dB

3.9 dB

NMF-NMF

4.9 dB

13.0 dB

6.6 dB

GIG(y|γ, ρ, τ ) =exp{(γ − 1) log y − ρy − τ /y}ρ√2τ γ/2 Kγ (2 ρτ )

γ/2

[16]

(A·2)

ͨͩ͠ Kγ ͸ୈ 2 छม‫ܗ‬ϕοηϧؔ਺Ͱ͋Δɽ·ͨɼMGIGC[17]

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IEEE TASLP, vol.22, pp.493–504, 2014.
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Yoshii, “A unified Bayesian model of time-frequency clustering and low-rank approximation for multi-channel source
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pp.177–180, 2003.
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982, 2013.
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pp.439–446, 2010.
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Ziehe, and A. Benichoux, “The 2011 signal separation evaluation campaign (SiSEC2011):-audio source separation,” Latent Variable Analysis and Signal Separation, pp.414–422,
Springer, 2012.
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1998.

ͷ֬཰ີ౓ؔ਺͸ҎԼͷൺྫࣜͰද͞ΕΔɽ

MGIGC (X|γ, R, U ) ∝ |X|γ−M exp{−tr(RX + U X −1 )}(A·3)ँࣙ ຊ‫ڀݚ‬ͷҰ෦͸ɼJSPS Պ‫ݚ‬අ 24220006ɼ15K12063 ͷࢧԉΛड͚ͨɽ

จ
[1]
[2]

[3]

[4]

[5]

[6]

[7]

[8]

‫ݙ‬

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pp.626–634, 1999.
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pp.1830–1847, 2004.
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expectation-maximization source separation and localization,” IEEE TASLP, vol.18, no.2, pp.382–394, 2010.

—7—