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1 6 / 6 ( ) / 53

2 6 ( ) 1 / 53

3 nodeedge 2 u v u v (u, v) 6 ( ) 2 / 53

4 Twitter 6 ( ) 3 / 53

5 Facebook 6 ( ) 4 / 53

6 N N N 1,2,..., N (i, j )i j 10 i j 2(i, j ) 1 6 ( ) 5 / 53

7 (u, v) u v 12 N N N 1,2,..., N (i, j )i j ( ) 6 / 53

8 A = A = ( ) 7 / 53

9 A = A = ( ) 8 / 53

10 6 ( ) 9 / 53

11 21 (u, v),(v, w),(w,u) 3 (u, v, w) 6 ( ) 10 / 53

12 i i 6 ( ) 11 / 53

13 A A n (i, j )i j n A 2 (i, j ) A i,1 A 1,j + A i,2 A 2,j A i,n A N,j k (i,k),(k, j )1i k j tr(a 3 )/6 = ((A 3 ) 1,1 + (A 3 ) 2,2 + + (A 3 ) N,N )/6 tr(a 3 )3 6 6 ( ) 12 / 53

14 i i A = UΛU T (Λ = diag(λ 1,λ 2,...,λ N )) tr(a 3 )/6 = 1 6 N λ 3 k k=1 i /2 1 2 (A3 ) i,i = 1 2 (UΛ3 U T ) i,i = 1 N λ 3 2 k U 2 i,k 95% 6 ( ) 13 / 53 k=1

15 Normalized Cut 6 ( ) 14 / 53

16 Cut Normalized Cut V 2 A,B A B = V, A B = wsum(a,b) = w uv u A,v B w uv uv w uv = 0 A B 1 6 ( ) 15 / 53

17 Normalized Cut Normalized Cut Normalized Cut wsum(a,b) wsum(a,v ) + wsum(a,b) wsum(b,v ) wsum(a, A) wsum(a,v ) + wsum(b,b) wsum(b,v ) Normalized CutNP 6 ( ) 16 / 53

18 Normalized Cut W d i = W i 1 +W i W i n D = diag(d 1,d 2,...,d n ) L W 2 x x i i A B Normalized Cut Shi Malik (2000) 6 ( ) 17 / 53

19 Normalized Cut x T Lx = w i j (x i x j ) 2 0 i <j ( ) 18 / 53

20 Normalized Cut 2Cut A n A B n B A n B B n A A n B B n A 6 ( ) 19 / 53

21 Normalized Cut Normalized Cut Normalized Cut ( ) 20 / 53

22 PageRank 6 ( ) 21 / 53

23 PageRank PageRank PageRank Web PageRank PageRank 6 ( ) 22 / 53

24 Web PageRank 1 Web 1 u v Web u Web v 6 ( ) 23 / 53

25 PageRank PageRank Web 1 Web = PageRank WebWeb Web PageRank 6 ( ) 24 / 53

26 PageRank (1) PageRank 6 ( ) 25 / 53

27 PageRank (2) PageRank 6 ( ) 26 / 53

28 PageRank (3) PageRank 6 ( ) 27 / 53

29 PageRank PageRank WebN N x t x t i t Webi i d i 1/d i Web A x 0 = (1,1,...,1) T, x t+1 = A T x t PageRank x x = A T x x PageRank A T 1 A 11 6 ( ) 28 / 53

30 PageRank A M n (R)λ k λ 1 > λ 2 λ 3 λ n x 0 x t+1 = Ax t / x t+1 = x t+1 x t+1 t x t λ 1 A v k x 0 = c 1 v 1 + c 2 v c n v n x t = const. (c ( 1 λ1 t v 1 + c 2 λ2 t v c n λn t v n) ( ) λ2 t ( ) ) λn t = const. c 1 v 1 + c 2 v c n v n λ 1 λ 1 6 ( ) 29 / 53

31 PageRank PageRank Web00 Web 0 PageRank0Web Perron Frobenius 6 ( ) 30 / 53

32 PageRank PageRank Web 1 WebWeb Web Web100α% Webα α/n (1 α)a +(α/n)e E 11 6 ( ) 31 / 53

33 HITS 6 ( ) 32 / 53

34 HITS (Hyperlink-Induced Topic Search) HITS HITSPageRankWeb HITS2 authority hubweb hub 6 ( ) 33 / 53

35 authorityhub (1) HITS Web authority Webhub WebhubWeb authority A PageRank authority x x = (1,1,...,1) T hub y y = (1,1,...,1) T 6 ( ) 34 / 53

36 authorityhub (2) HITS x A T y, x x/ x, y Ax y y/ y () x, y x = const. A T y, y = const. Ax λx = A T Ax, λy = AA T y x A authority y A hub 6 ( ) 35 / 53

37 6 ( ) 36 / 53

38 (Latent Semantic Analysis, LSA) (Latent Semantic Indexing, LSI) Web A(i, j )Webi j 6 ( ) 37 / 53

39 1: () 2: () 3: () A = () 4: () 5: () : : 12,1,1,1,1 6 ( ) 38 / 53

40 A A = UDV T U = D = diag(2.65, 2.14, 0.893, 0.662, 0.422) V T = ( ) 39 / 53

41 A i a i = k σ k V ki u k σ k 0a i A UDV T U,V k + 1 q A T q U T DV T i U T q : q = (1,0,1,0,0,0) T k = 1(DV T ) T U T q = (1.68,1.46,0.62,0.73,1.25) T k = 2(DV T ) T U T q = (1.68,1.23,0.97,1.17,1.08) T k = 5(DV T ) T U T q = (2,1,1,1,1) T 6 ( ) 40 / 53

42 6 ( ) 41 / 53

43 w h Z wh 2 x, y Z wh x T y wh N N 6 ( ) 42 / 53

44 A = UΣV T r rank A Ã A ΣΣσ r +1,σ r +2,...0 Ã = U ΣV T A Ã A M m,n (R) A F = m n ( ) Ai,j 2 = tr A T A = i =1 j =1 k σ 2 k 6 ( ) 43 / 53

45 j i A i,j 2 k (i, j ) A i,j,k 6 ( ) 44 / 53

46 6 ( ) 45 / 53

47 i j 515 A i,j A i,j = 0 A i,j = 0 A i,j U i T j 6 ( ) 46 / 53

48 k j v j = (V 1j,V 2j,...,V k j ) T i u i = (U 1i,U 2i,...,U ki ) T i j u i Tv j = U 1i V 1j + +U ki V k j ( ) 2 minimize Ai j B i j A i,j 0 subject to rank B k B = U DV T D k 6 ( ) 47 / 53

49 EM B minimize ( ) 2 Ai j B i j i,j subject to rank B k B i j 6 ( ) 48 / 53

50 6 ( ) 49 / 53

51 6 ( ) 50 / 53

52 PARAFAC (parallel factor analysis) l m n A A i j k, 1 i l, 1 j m, 1 k n Q A i j k = d q x i y j z k q=1 Q NP Q l,m,n 6 ( ) 51 / 53

53 Tucker l m n A A i j k, 1 i l, 1 j m, 1 k n Q R S A i j k = d qr s X i q Y j r Z ks q=1 r =1 s=1 X T X = I, Y T Y = I, Z T Z = I 6 ( ) 52 / 53

54 Tucker minimize ) 2 (A i j k B i j k i,j,k subject to B i j k = Q R S q=1 r =1 s=1 d qr s X i q Y j r Z ks X T X = I, Y T Y = I, Z T Z = I Higher Order SVD A l mn A m nl A n lm 6 ( ) 53 / 53

ver Web

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