非負値行列因子分解 NMF の基礎 ... - kecl.ntt.co.jp

1 9 _02_ _ .mcd Page 112/08/10 09:24 NMF Nonnegative Matrix Factorization and Its Applications to Data/Signal Analysis NMF:Nonnegative Matrix Factorization NMF NMF NMF NMF(Nonnegative Matrix Factorization) 0 1(a) 1(b) NMF.

2 Nonnegative MatrixFactorization (1),(2) 0 (3) (4) NMF NMF NMF NMF NMF NMF NMF $ NMF 2 I J X NMF I K T K J V K NMF NMF 829 NTT (NTTC ommunicationScienceLaboratories.)

3 NIPPONTELEGRAPH ANDTELEPHONE CORPORATION,Kyoto-fu,619-0237 Japan). 2012 9 2012 / 9 _02_ _ .mcd Page 212/08/10 09:24 t v 0 t =[t , ,t ] v =[v , ,v ] t v = t v 2 0 X TV D(X,TV) NMF Eu:Euclid 2 KL Kullback-Leiblerdivergence,IS:Itakura-Sa itodiver-gence(5) 3 D (X,TV)= d ( ,t v ) d d ( ,t v )=( t v ) d ( ,t v )= log t v +t v d ( ,t v )= t v log t v 1 3 =1 Euclid 0 KL diver-gence IS divergence IS divergence t v d (9,10)=d (900,1000) IS divergence NMF K 4 K D(X,TV)

4 NMF K 4 NMF K=7 = NMF =. Multiplicative update rulesD (X,TV) NMF Multiplicativeupdate rules (i,j) =t v 3 , ,2012830 $ NMF = =1 = Euclid Eu KL divergence KL IS divergence IS N NMF K X Euclid z 9 _02_ _ .mcd Page 312/08/10 09:24 t t v v v v t t ( )KL divergencet t v v v v t t ( )IS divergencet t v v v v t t ( ) (6) T,V T V update rule Multiplicative t v T,V 0 i,j = 1 =.

5 $ NMF D (X,TV) T V F (T,V) 3 F (T,V)= [(t v ) 2 t v ]F (T,V)= (t v logt v )F (T,V)= t v +logt v F F R 1 F(T,V) F (T,V,R) 2 R F(T,V)=minRF (T,V,R) F (T,V,R) R F (T,V,R) T V 5 T V R R T V R NMF 3 R U r >0 r =1 u >0 F (T,V,R)= (t v ) r 2 t v F (T,V,R)= t v r logt v r F (T,V,R,U)= r t v +logu +t v u u Euclid F KL divergence,IS divergence F 1 2 Jensen R r =1 L(T,V,R, )

6 =F + ( r 1) NMF 831 k 9 _02_ _ .mcd Page 412/08/10 09:24 r 0 L r = (t v ) r + =0 r =t v k =(t v ) r =t v t v =t v ( ) F F F 1 2 T V t v 0 F t =2t v r 2 v =0 F v =2v t r 2 t =0 t = v v r v = t t r ( ) ( ) N N. 20 News-groups( 1) X NMF stopwords( 2) documentfrequency I=60,835 J=18,774 0 150 I J X K=20 KL di-vergence NMF T,V ( )

7 50 Matlab IntelCore i7 965 145 6 20 NMF 8 T NMF 2 baseball 4 17 V NMF 6 N.$ SignalSeparationEvaluationCampaign , ,2012832 x $/ Newsgroups NMF (a) T (b) V z ( 1) ( 2) 9 _02_ _ .mcd Page 512/08/10 09:24 SiSEC ( 3) nine_inch_nails-the_good_soldier 20 10 10 T convolutive NMF(7) IS divergence 10 K=10 20 K=30 NMF 1 10 11 30 7 (a) V 1 10 20 11 30 10 (b) (c) (b) (c)

8 K NMF NMF NMF (8) ( ) N. NMF LDA:LatentDirechletAllocation(10) LDA ( ) Lee and Seung, Learning the parts of objects withnonnegative matrix factorization, Nature, vol. 401, pp. 788-791, 1999.( )A. Cichocki, R. Zdunek, Phan, and S. Amari. Nonnegative Matrixand Tensor Factorizations : Applications to Exploratory Multi-wayData Analysis and Blind Source Separation, Wiley, 2009.

9 ( )W. Xu, X. Liu, and Y. Gong, Document clustering based on non-negative matrix factorization, In Proc. ACM SIGIR, pp. 267-273,2003.( )P. Smaragdis and Brown, Non-negative matrix factorization forpolyphonic music transcription, In Proc. WASPAA 2003, pp. 177-180, Oct. 2003.( )C. F votte, N. Bertin, and J-L. Durrieu, Nonnegative matrixfactorization with the Itakura-Saito divergence : With application tomusic analysis, Neural Comput., vol. 21, no. 3, pp. 793-830, 2009.( )M. Nakano, H. Kameoka, J. Le Roux, Y. Kitano, N. Ono, and , Convergence-guaranteed multiplicative algorithms fornon-negative matrix factorization with beta-divergence, In 2010, pp.

10 283-288, Aug. 2010.( )P. Smaragdis Convolutive speech bases and their application tosupervised speech separation, IEEE Trans. Audio, Speech, andLanguage Processing, vol. 15, pp. 1-12, 2007.( )H. Sawada, H. Kameoka, S. Araki, and N. Ueda, Efficient algorithmsfor multichannel extensions of Itakura-Saito nonnegative matrixfactorization, In Proc. ICASSP 2012, pp. 261-264, March 2012.( ) vol. 51, no. 9,2012.( z) Blei, Ng, and Jordan, Latent direchlet allocation, Learn. Res., vol. 3, pp. 993-1022, 2003. 24 4 27 24 5 14 3 5 NTT VLSI CAD 12 21 13 IEEE NMF 833 NMF (a) V z z (b)NMF z (c)NMF z ( 3)