title: Classification of signals and graphs by algebraic spectral approaches. Signal and Image processing. Université de Bretagne occidentale - Brest, 2018. English. &#x27E8;NNT : 2018BRES0107&#x27E9;. &#x27E8;tel-03669985&#x27E9;
text: Classification of signals and graphs by algebraic spectral approaches
  
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TV TV EA EL (, EL A L, ) A L C L D C R2 X G R3 F G0 4 G 12 ⇥ 12 TV EA EL C10 H10 N2 O2 C6 H2 Cl4 ↵ A L (M M ) ↵ 10 10 64 34 x, x0 2 H ± n 2n(n 1)/2 ± ± N A L D ⇡ A L L (A) (DG ) | L | (L) L A (L = D A) D A L A L A A L (A) ) • ( ) (L) • • • A L A L A L ) ( (,, ) ) ) A (L = D L A) ) (A) ) A L ↵ (L) ) A ) ) VPV ) (DGA) L 1 G = (V, E, W) V = {v1, v2,., vn } n =| V | (vi, vj ) E = {eij } = {(vi, vj ) | i, j = 1, 2,. n; i 6= j} ✓ V ⇥ V eij vi m =| E | |.| vj i N (i) = {j 2 V : (i, j) 2 E} E!R W wij i j w wij eij wij i j wij 2 {0, 1} G v1 0.6 v2 0.75 v3 G = (V, E) G 0.1 (i, j) 2 E 0.8 v4 (i, j) 2 E (j, i) 2 E wij = wji G (i, j) 2 E d(vi ) G 0 0 2 v1 v2 v3 v4 d(vi ) = 2 0 0 n G Kn Kn K4 n ⇥ (n 1)/2 (n 6 1) 3 v1 v4 v2 v3 K4 vi 2 V vi0 2 V 0 V v10 v1 v20 v2 v30 v3 n=6 m=9 (vi, vi+1,., v j ) vj vi vj V0 vi v1 v3 v1 v2 v3 v4 v1 x:V!R x 2 Rn⇥n v1 v2 v3 v4 G x x = (x(v1 ), x(v2 ),., x(vn )) = (x1, x2,. , xn ) v3 G I (G) = n X d(vi ) i2V I (G) = 0 G G I (G)! 0 M = MT M 2m n M M 2 Rn⇥n M M T race(M) = n X M M = [mij ] M mij i=1 n⇥n M M M = U ^ UT ^ 2 Rn⇥n ^ U 2 Rn⇥n U M ^= ( 1, 2,., n ), Mui = U = [u1 | u2 |. | un ] i ui UT = U U M Mu1 = ( 1, 2,., 1 u1, Mu2 = 1 U n) 2 u2, M 1 6= 2 =) u1.u2 = 0 M n M M n⇥n 1, EM = n X EM G 1, 1 k 2 N⇤ M n T race(M) n i i=1 M 2,., n k 2..., n G Mk = M (G) = n X i=1 | i |k 2,., n Mk k Mk G k=1 G G A = [aij ] = A A(G) 8 > < 1 aij (vi, vj ) 2 E > : 0 (0, 1) vi ⇠ vj G v1 v2 v3 v4 A 0 0 B B B 1 B A=B B B 1 @ 0 1 1 0 1 1 0 1 0 0 1 C C 1 C C C C 0 C A 0 G vi (n ⇥ m) C = {cij } ek vi 0 0 k vj 0 0 0 cik = cjk = 1 C C = [cij ] = 8 > < 1 (eij ) > : 0 v1 v2 v3 v4 C 0 1 B B B 1 B C=B B B 0 @ 0 1 0 0 1 1 1 0 0 vi 0 1 C C 1 C C C C 0 C A 1 vi d(vi ) = X d(vi ) aij j2i vi D d(vi ) G 0 d(v1 ) B B B 0 B D=B B B @ 0 0 ··· d(v2 ) ··· 0 ··· G 0 1 C C 0 C C C= C C A d(vn ) (d(v1 ), d(v2 ),. , d(vn )) 0 2 B B B 0 B D=B B B 0 @ 0 0 0 3 0 0 2 0 0 0 1 C C 0 C C C C 0 C A 1 A D d(vi ) = n X wij j=1 W = [wij ] 2 Rn⇥n W = D 1A x y = Lx yi yi = [Lx]i = X wij (xi xj ) j2N (i) yi i G L x 2 Rn n⇥n L xT Lx = 1 X wij (x(i) 2 x(j))2 (i,j)2E 0, 8x x xT Lx x G L=D A A = [lij ] = D 8 > > > > < 1 d(vi ) > > > > : 0 i 6= j and vi ⇠ vj i=j v1 v2 v3 v4 L 0 B B B B L=B B B @ L 2 1 1 1 3 1 1 1 2 0 1 0 G 0 1 C C 1 C C C C 0 C A 1 [1, 1,., 1]T ”0” L G G L LN LN = D 1/2 LD 1/2 = I G D 1/2 AD 1/2 A L I D 1/2 1 D 1/2 (i, i) = p d(vi ) 8 > > 1 > > > < 1 p LN = [lij ]N = > d(vi ).d(vj ) > > > > : 0 0 2 i = j ; d(vi ) 6= 0 if vi ⇠ vj G G G G i 2 [0, 2] u i i 2 [0, 1] 2]1, 2] x̂ G x x =< u, x >= n X i=1 ) ) x(i)ui G |L| = D + A L n X i=1 n X i = 2m 2 i = 2m + n X i=1 i=1 1, L L d2i 2,., n 2m n X i = 2m i=1 m>0 2m/n G LE(G) n X i n X1 p i LE(G) = EL (G) = i=1 LE(G) n LE(G) LEL(G) = i=1 2m n LEL(G) LE(G) LEL(G) μ 1, μ2,. n G 1 2m n 1 G n X n X i = 0; 2 i = 2m; A= i i=1 i=1 i=1 n Y G i 6= 0, 8i 2 {1, 2,., n} A=0 A 6= 0 max dmax G dave G A max dave  1  max G 1 G max max G G A EA (G) = n X i=1 | i | A EA (G) G ⇡ v v y = Ax; y(i) = X x(i) i⇠j A xT Ax = X x(i)x(j) eij E(G) M = [mij ] X mij xi xj xT Mx i,j X = xT x x2 i i M T T M = T M T = T T = n⇥n 1 2,..., n M xT Mx xT x x M 1 = max x n⇥n M 1, xT Mx xT x 2,., 1, { i }ni=1 n i = i = 2,., { i }ni=1 xT Mx x? 1,..., i 1 xT x xT Mx arg min x? 1,..., i 1 xT x min M M n⇥n k = 1, xT Mx = x2S xT x minn max S✓R =k x6=0 1 = max x2Rn xT Mx xT x 2,., xT Mx x2T xT x maxn min n = minn x2R xT Mx xT x T ✓R =n k+1 x6=0 n n k k = 1 S S 1, 2,., T k 2,., 1, n 1 = 2 = k, k+1,., 2,., n xT Mx x6=0 xT x min xT Mx = min ||x||=1 xT Mx xT Mx = min T x6=0 xT x ||x||=1 x x min x? n = k 6= 1 n⇥n M 1, Rn T 1 x? 1 1 xT Mx x6=0 xT x max xT Mx = max ||x||=1 G = (V, E) L 1 = 0, 1 = [1, 1,.., 1] 2 = min x6=0 x? 1 T xT Lx = min xT x Px6=0 X (i,j)2E i xi =0 = max xT Lx = max x6=0 x6=0 xj ) 2 (xi X x2i i2V X (xi xj ) 2 (i,j)2E X x2i i2V x 2 x G G n 0= 1  2 ,. ,  n 2 =0 2 2 G G1 0 1 0C A L2 BL 1 L=@ 0 L G2 L1 L2 L L G 2 2 L yT Ly = P (y(u) y(v))2 y u⇠v v y(v) P (u, v) u⇠v u v Rayleigh quotient ( k, k) L kM kM M J kM Jk Jk Jk⇤ kZk = kZk = kZk⇤ = kZxk max x sX kxk zij i,j (Z) k i Bi k M̃k = n X k M i i Bi i=k kM k Jk M M n⇥n M G Bi M kX s 1 M= i Bi n X + i=1 ū= ū 1  ũ | {z i Bi i=ks } ũ= | {z } ks M = LN i 2 [0, 2] ks = arg max [ i] {1in| 0 i 1} M X M= ū= | i 1 {z i Bi } + X 1< i 2 | ũ= {z i Bi } G Â 2 ,.,  n kd = arg minkA 1kn n X i=k A(i, k) 2 {0, 1} i Bi k Ĥ = n X i Bi i=kd Ĥ Â(i, j) = 0 Â(i, j) = 1 1 2 1 Ĥ(i, j) > 2 Ĥ(i, j) < A F̂k = n X i=k i Bi k 2 {1, 2,. j=1 (x) = [ p 1 1 (x), · · ·, p j j (x), · · · ]. ”k” p(t+k) = p(t) Pk = p(t) (D 1 A)k. [(D 1 A)k ]ij vj vi p(0) k p(k) p(k) = (D 1 A)k p(0) k p(k) k vj G G⇥ A⇥ j th G0 G0 G G(V, E) G0 (V 0, E 0 ) G⇥ V⇥ = {(v, v 0 ) 2 V ⇥ V 0 |v 2 V, v 0 2 V 0 } E⇥ = {((u, u0 ), (v, v 0 )) 2 V⇥ ⇥ V⇥ |(u, v) 2 E, (u0, v 0 ) 2 E 0 } G⇥ G G G G p G0 G⇥ G0 A G0 A0 A⇥ = A ⌦ A0 G⇥ G0 G⇥ G p K⇥ (G, G0 ) = G0 |V⇥ | h p X X i,j=1 l=0 1 l l (D⇥ A⇥ ) i ij (⌦) p2N ⌦ = G0 G G ⊗ G0 G0 G G0 G V⇥ D⇥ = D ⌦ D0 G⇥ 0, p 1, 2,..., (D⇥1 A⇥ )0 = I p 1 random walk kernel (K⇥ (G, G0 )) 1 l l p K⇥ |V⇥ | h 1 X X KGeo (G, G0 ) = i,j=1 l=0 l (D⇥1 A⇥ )l i p!1 |V⇥ | ij = X⇥ i,j=1 (I ⇤ Ã⇥ ) 1 ij Ã⇥ = D⇥1 A⇥ I G⇥ (I 0 l!1 Ã⇥ ) x = 0 (I Ã⇥ ) ( Ã⇥ )l x = x, 8 l 2 N x x 1 P Ã⇥ )(I+ Ã⇥ + 2 Ã2⇥ +...) l=0 (I + Ã⇥ + ( Ã⇥ )2 +... + ( Ã⇥ )1 ) Ã⇥ (I ( Ã⇥ )l l Ãl⇥ = ( I Ã⇥ ) 1 μ⇥,M ax 1\ KGeo < 1 \ μ⇥,M ax G(V, E) G GH (VH, EH ) GH v G (v, w) 2 EH ΩH : V H! V (ΩH (v), ΩH (w)) 2 E Ω = {ΩH |GH v G} GH GH |Ω| G #{ΩH |GH v G} p 2 {3, 4, 5} Gp = {Gi (Vi, Ei )|i = 1, 2,..., M } N p p G G fG M Gi #{Ωi |Gi v G} G fG G p G0 G N, N 0 p Kg T 0 0 Kg (G, G0 ) = hfG, fG 0 i = f G f G0. Kg p p 3, 4, N (N p ) 5 p X := {Xj }m j=1 S = {1, 2,..., a} Xj ⌘ (Xj ⇠ ⌘) ⌘ m m 1 X ⌘ ̃ (i) = (Xj = i), m j=1 m i2S (Xj ⇠ ⌘) X := {Xj }m j=1 S = {1, 2,..., a} ⌘ ⇠>0 >0 m= P {k ⌘ ⌘ ̃m k1 2(log 2· a + log ( 1 )) ⇠2 ⇠} . (vi,vj )2E (vi0,vj0 )2E 0 W X X 0 KEdge 0 (`(vi ), ` (vi0 )) (`(vj ), ` (vj0 )) KW (W(vi, vj ), W(vi0, vj0 )), (vi,vj )2E (vi0,vj0 )2E 0 KW (h) 0 (` (vi0 ), ` (vj0 )) KW L edge (G, G0 ) = KEdge (G0, G0 0 ) + KEdge (G1, G0 1 ) + · · · + KEdge (Gh, G0 h ). 3 (T V ) (EL ) TV C EL SV M (T V ) f f [a, b] f x 7! f (x) x 2 [a, b] [a, b] ⇢ R TV TV `1 TV f T V (f ) = +1 Z | f 0 (t) | dt 1 f0 f f f 0 (tp ) = 0 {tp : p = 1,..., M } T V (f ) = M M X1 | f (tp+1 ) f (tp ) | p=1 f (t) = exp( t2 ) TV f (t) = sin(⇡t)/(⇡t) (T V (f ) = 2) tp 2 [p, p + 1] (T V (f ) = +1) | f (tp+1 ) f (tp ) |⇠| p | 1 h T V (f ) = lim h!0 +1 Z | f (t) h) | f (t |h| dt. 1 f (T V (f ) < +1) f0 f fˆ0 (!) = i! fˆ(!) f |! || fˆ(!) |6 +1 Z | f 0 (t) | dt = T V (f ). 1 | fˆ(!) |6 T V (f ). |!| | fˆ(!) |= O(| T V (f ) ||! | 1 ) f f (t) = sin(⇡t)/(⇡t) fˆ = 1[ ⇡,⇡] | fˆ(!) |6 ⇡ |! | 1 [ ⇡, ⇡] k f kV = +1 | fˆ(!) | f fM [n] f 1/M TV h = 1/M T V (fM ) = X | fM [n] fM [n 1] |. n T V (fM ) fM = [fM (0), fM (1),..., fM (M 1)] 2 RM fM T V (fM ) =k rfM k1, k · k1 `1 r 0 1 B B B B r=B B B @ (M 1 1 1 1 1 1) ⇥ M 1 C C C C C, C C A M r f (x1, x2 ) T V (f ) = +1 Z +1 Z 1 1! | rf (x1, x2 ) | dx1 dx2,! r(·) ⇣ @f (x, x ) 2! @f (x1, x2 ) 2 ⌘1/2 1 2 +. | rf (x1, x2 ) |= @x1 @x2 TV | ∆h f (x1, x2 ) |= h ⇣ f (x, x ) 1 2 f (x1 h h, x2 ) 2 + f (x1, x2 ) f (x1, x2 h! | rf (x1, x2 ) | | ∆h f (x1, x2 ) | T V (f ) 6 lim h!0 +1 Z +1 Z 1 1 | ∆h f (x1, x2 ) | dx1 dx2 6 h) 2 ⌘1/2 p 2 T V (f ). , f (x1, x2 ) fN [n1, n2 ] f (x1, x2 ) N N T V (fN ) = 1 XX⇣ fN [n1, n2 ] N n1 n2 fN [n1 1, n2 ] 2 + fN [n1, n2 ] fN [n1, n2 1] 2 ⌘1/2. TV G(V, A, f ) fn (vm 2 Nn ) f ̃n vn f ̃n = X m2Nn An,m fm, f̃ = [f ̃0,..., f ̃N Nn N 1] T = A f, vn A f̃ f f G(V, A, f ) T VG (f ) =k f A f k1, G A f = f̃ A f `1 f fn = fn mod N T V (f ) = N X1 | fn fn 1 mod N | n=0 (vN N ⇥N 1, v0 ) (fN = f0 ) N k · k1 TV TV TV TV TV TV TV x̂ = nZ x x(t))2 dt + T V (x) (y(t) Ω o T V (x) x Ω y(n) w(n) y(n) = x(n) + w(n) y(n) N x(n) x̂ = x n 1 NX1 2 n=0 2 | y(n) x(n) | dt + N X1 | x(n) n=1 x(n o 1) |. x(n) y(n) x(n) N x rx TV x̂ = x n1 2 ky o x k22 + k rx k1. k rx k1 TV A A m m vm m A G(V, A, f ) v = v0, v1,..., vR 1 A ir = vr 8 > > <0, r=0 > > :1, 1  r  R, A A Avr = vr + ir vr 1. G(V, A, f ) f vr f G vr Anorm vr k1 T VG (vr ) =
  
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  | max | Avr Anorm 1 A (A/ | max |) vr T VG (vr ) = vr vr A | max | vr ir | max | r = 0 v = vr = v0 vr 1 1. i0 = 0 v T VG (v) = 1 | max | k v k1. A v (k v k1 = 1) T VG (v) = 1 | max | 1+ | max |  2. 2 0 G m, n 2R A m < vm vn n T VG (vm ) > T VG (vn ), 0 > 1 > ··· > M m, M 1 0 1 n 2 C A vm vn T VG (vm ) < T VG (vn ), | m | | max | n max | max | /| max | x(t) | X(f ) |2 X(f ) | X(f ) |2 x(t) Ex Ex x(t)? Ex = hx(t), x (t)i = +1 Z 2 | x(t) | dt = 1 +1 Z | X(f ) |2 df, 1 x(t) X(f ) Ex x(t) Ex x(t) | x(t) |2 | X(f ) |2 X(f ) Ex x(t) x(t
  
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  x(t) |2 ΨC x(t) ΨC [x(t)] = dx(t) dt = ẋ2 (t) ẋ(t)!2 d2 x(t) x(t) dt2! x(t) ẍ(t), ẍ(t) x(t) t ΨC x(t) ΨC [x(t)] x(t) 2 6 ẋ(t) ΨC [x(t)] = 4 ẍ(t) 3 x(t) 7 5. ẋ(t) HΨ = EΨ H E Ψ (E, Ψ) ⇡ H = ↵I + A A ↵ H H / (A) A 0, 1, 2⇡ 2p E>0 ⇡ (E < 0) ⇡ ⇡ ⇡ ( p d) ⇡ s p p π σ E HΨ = EΨ H = ↵I + A (T race(A) = 0) ⇡ EA (G) = n X i=1 | μi |, n ith μ i ⇡ Ei G A EA (G) G E⇡ = ↵ne + EA (G) ⇡ ne ↵ E⇡ ⇡ E⇡ EA (G) ⇡ G n X μi = 0, i=1 m n X n n X μi 2 = 2m, i=1 n X i = 2m, i=1 k 2 = 2m + i=1 μ1 > μ2 > · · · > μn (L) i (k > 1) (d(vi ))2, vi 2V (A) d(vi ) X 1 > 2 > ··· > n vi n i = 0 i 2 {0, 1,..., k 1} n k > 0 EA • EA (G) > 0 • G • G m=0 G1 G2 EA (G) = EA (G1 ) + EA (G2 ) G1 EA (G) = EA (G1) EA (L) EA EA EL (G) = n X i i=1 G i 2m L i = i 2m n i 2 {1, 2,..., n} 2m, n 2m/n L 2m/n n X i = 0, i=1 n X n 1 X⇣ M =m+ d(vi ) 2 i=1 2 i = 2M i=1 2m ⌘2. n | L |= D + A L ⌫1 > ⌫2 > · · · > ⌫n G n X E|L| (G) = ⌫i i=1 2m. n DG vi DG kj DG ij vj j vk (1) DG EA E DG = n X | ⌘i |, i=1 DG ⌘1 > ⌘2 > · · · > ⌘n 1 > 2 > ··· > n > 0 Mn (C) B 2 Mn (C) k B k? = B T race(B) ⇠1 n⇥n ⇣ ⇠1,..., ⇠ n T race(B),..., n B ⇠n T race(B) n T race(B) In n? = n X i=1 ⇠i B B 2 Mn (C) ⌘ T race(B) In n B B G EB = n X i=1 i=1 i T race(B) n B A Pn | ⇠i |=k A k?, T race(B) In n T race(A) = 0 B L G 2m (tr(L) = 2m) m EB = n X i=1 n⇥n ⇠i T race(L) In n 2m = L n.? Mn (C) B EA EA q n 2m + n(n 1) | det(A) |2/n 6 EA 6 m p 2mn, A n m ⇡ EA = p 2mn ⇡ G ⇡ 1 EA p 2 m 6 EA 6 2m, p EA > 2 n 1, m n p EA = 2 m G G G EA = 2m G 1 EA ⇡ G n m n m 2m > n 2m EA 6 + n r (n h 1) 2m ⇣ 2m ⌘2 i n n 2 K2, Kn G q Kn K2 2 ( 2m n ) )/(n 1) n 2 K2 n n 2 (2m 2 ⇡ EA G n>2 p p n( n + 2) p EA 6 8 n = 2⌫ ⇣ p ⌘ p 2 ⌫, ⌫+2 ⌫, ⌫+24 ⌫ G t EA n = 64,
  
  256, 1024, 4096,... G EA 6 G p p 2 n)/4, (n + 2 n)/4) n p n (1 + n) 2 (n, (n + p n)/2, (n + k G n (n, k,, μ) >0 μ>0 G G x2 + (μ )x + (μ μ=0 k r, s k) = 0 G n 6 m 6 2n n 4 q p EA > 2 m + 2 (m n + 2)(2n m 4), G | V1 |= 2 | V2 |= n 2 V1 G V2 V1 V2 V1 EA m (m n 2) V2 EL EL EL 6 p 2M n, 1 X M =m+ 2 v u 2m u + t(n EL 6 n d(vi ) vi 2V " 2m n 1) 2M!2 # 2m n!2,, p 2 M 6 EL 6 2M. (n, m) G k ↵(k 2) 0 ↵ ↵ > 1, k > 2 k=2 G G (n, m) p G (p > 1) v u u 2m EL 6 p + t(n n p=1 p [2m (2m/n)2 ]/(n ↵ > 1, k > 2 ↵(k 1) " 2m p n p) 2M!2 # G 0, 1 p = n G n EL = 0 k ↵(k k=2 G ↵ 2) = p p = n/2 p EL = 2 M (n, m), 1 p 2) G Kn/2,n/2 EL = 2M G EA EL n B i (B), i 2 {1, 2,..., n} ⇢i (B), i 2 {1, 2,..., n} i (B) =| ⇢i (B) | i 2 {1, 2,..., n} A B1 B2 n (2 Mn (R)) B B1 + B2 = B n X i=1 i (B1 ) + n X i=1 i (B2 ) > n X i (B), i=1 P PB1 PB2 G1, G2 G E A 1 + E A 2 > EA A1 + A2 = A G Ḡ n EA + EĀ > 2(n G⇠ = Kn 1), G⇠ = K̄n G EA 6 2m p 2(∆ n m S∆+1 m ∆), G n 2m + ∆ ∆ ∆ 1 EL EA G n m d1, d2,..., dn E L 6 EA + n X 2m, n di i=1 2m/n G L 2m In = D n EL = 6 n X i=1 n X i ⇣ i ( A) + 6 i=1 i (A) + i=1 6 EA + D 2m n In ⌘ n X i=1 n X i=1 ⇣ n X di i 2m i In. n 2m i⌘ In, n h A+ D i=1 n X h A+ D 2m In = n A i ⇣h ⇣h D D 2m i⌘ In, n 2m i⌘ In, n 2m, n ⇣ di 2m n ⌘, i 2 {1, 2,..., n} EL > EA G n m d1, d2,..., dn ( max EA, n X i=1 di 2m n 2m/n ) 6 E L 6 EA + d i 2m. n G EL EL p p EL EA ( (,, ) ) (Gi, Gj ) = ↵ ⇥ (Gi, Gj ) + (1 ↵) ⇥ (Gi, Gj ), ↵ 2 [0, 1] (Gi Gj ) Ki,j = K S(Gi, Gj ) 1 X 1 n! n=0 ( S(Gi, Gj ))n = exp ( M ⇥M (, ( • S(Gi, Gj )),, ),, ) • ⇡ • (,, ) SSE ↵ SSE SSE C↵ SEE SSE C↵ • C↵ (C↵ ) & & 1178 10 1 9 ± & ä [ ±
  
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  ±0.21 ±0.87 ±0.07 ±1.09 ±0.28 ±0.94 ±0.95 ±0.83 ±0.38 ±0.27 ±0.18 ±0.94 ±0.47 ±0.49 ±0.07 ±0.08 ±1.20 ±0.12 ±1.95 ±0.30 ±0.20 ±2.04 ±0.70 ] p [ ] [ ] [ ] [ ] & ä [ 0.03600 1700 17.400 0.400 1.400 0.1200 20.600 21.800 100 20 1900 0.1800 5100 51.200 1.600 20 2100 400 600 81 81 40 4200 5 5 100 4 1200 9 9 120 1900 48 300 10 2700 10 2700 2500 300 2100 300 10 500 5800 1100 3 38 103 ] p
  
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  ] [ ] [ ] [ ] ( ) ( ) (A, L,..., | L |) 4 (A) | L | (DG ) (L) L A A L L A ( A A ) L L A A L L ρ n |Ψi i |Ψi i ρ pi {|Ψi i}ni=1 H ρ= n X i=1 pi i pi |Ψi ihΨi |. H ρ ρ T race(ρ) = 1 ρ>0 ρ = ρ† D(H) ρ 2 D(H)) ρ L n Ψ(x, t) Ψ⇤ (x, t)Ψ(x, t) x m0 t (x1, x2, x3 ) Ĥ = ~2 2 r 0 {z } | 2m r2 = + @2 @xi @xi ~ ĤΨ = EΨ, V (x) | {z }, Ĥ E Ψ Ψ r2 Ψ + ~2 (E 2m0 V (x))Ψ = 0, V (x) = D L=D A L= ~2 2 r, 2m0 Ĥ = A. L N T L 2 ~ 2 2m0 r V (x) A I Ĥ = I Ã = L̃ Ĥ |Ψi i = L̃ |Ψi i = Ei |Ψi i ρ [Ĥ, ρ] = [L̃, L̃ ]=0 |V | ρ 1927 ρ ρ 2 D(H)) T race(B) S(ρ) = KB T race(ρ log ρ), B KB ρ G G G ρ= P L, d(vi ) i 1 2,..., n = 0 ρ A L S(ρ) = n X i log2 i i=1 G 0 log2 0 = 0 G 1 A 0 L G A L A L A ρ G ρ0 ρ⇤ ρ0 ρ = ρ0 + ⇠ ρ⇤ S(ρ) = S(ρ0 ) + ⌘Diagonal (ρ⇤ ) + ⌘Off ⌘Diagonal (ρ⇤ ) = ⇠ X ⇢⇤ nn log2 ̃ n n ⌘Off ⇤ Diagonal (ρ ), 2 1 2 X (⇢⇤ nn ) + o(⇠ 2 ) ⇠ ̃n 2 n X X |⇢⇤ |2 nm (ρ ) = ⇠ log2 ̃ n + o(⇠ 2 ), Diagonal ̃n ̃m
  
  ⇤ 2 n m6=n ⇠ ̃ n 6= ̃ m 8 n 6= m ρ0 ρ ρ = ρT. T r(ρ) = 1 ̃ (t) = ̃ n + ⇠ ̃ (1) + ⇠ 2 ̃ (2) + o(⇠ 2 ), n n ̃ (1) = ⇢⇤ n, nn n ̃ (2) = n X |⇢⇤ |2 nm. ̃n ̃m m6=n S(ρ) = X T r(ρ log2 ρ) ⇡ ̃ (t) log ̃ (t). 2 n n n S(ρ) S(ρ) ⇡ S(ρ0 ) + ⇠ dS(ρ0 ) 1 2 d2 S(ρ0 ) + ⇠ + o(⇠ 2 ). d⇠ 2 d⇠ 2 S(ρ0 ) dS(ρ0 ) = d⇠ d2 S(ρ0 ) = d⇠ 2 S(ρ) ⇡ S(ρ0 ) X n X (⇢⇤ nn ) 2 2 ̃n n ⇠ ⇢⇤ nn log2 ̃ n, X X + n 2 2 1 2 X (⇢⇤ nn ) ⇠ ̃n 2 n {z } ⇢⇤ nn log2 ̃ n n | ̃ (2) log ̃ n. n ⌘Diagonal X X |⇢⇤ |2 nm log2 ̃ n +o(⇠ 2 ). ⇠2 ̃n ̃m n m6=n | {z } ⌘Off Diagonal ⌘Diagonal A L=D ⌘Off Diagonal A A L D A L A L A A L A L |L| 3.8% 9% 21% 11 σa (A) = σb (A) = [−1.9, −1, −1, 0.2, 1, 2.7] σa (L) = [0, 1, 1, 3, 3, 6] σb (L) = A [0, 0.5, 1.2, 3.4, 4, 4.7] σ(B) B A L L A σa (L) = σb (L) = [0, 0.7, 2, 3, 3, 5.2] σa (A) = [−2.1, −1, −0.5, 0, 1.2, 2.5] L σb (A) = [−2.5, −0.7, 0, 0, 0.7, 2.5] B σ(B) 2n(n 1)/2 n (n) A L A L |L| L G |L| A (A, L ) 2 G (A, L ) A L L 1i μ1i μ2i 2i L A A L A (, )=↵ A(, ) + (1 ↵) L L(, k X (μ1i ) ↵ 2 [0, 1] L(, )= k X i=1 ( 1i 2 2i ), A(, )= i=1 μ2i ) 2 A = min( 1, 2) 1, 2 ↵ A L Graph & Vi graph (ta, ya ) (tb, yb ) (tc, yc ) yc < yb + (ya wab = arctan yb ) ✓ tb tb yb tb wab tc. ta ya ta ◆. (A)
  
  
  
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  600 12% 8% 6% ± & A ±0.22 ±0.14 ±0.28 ±1.15 ±0.14 L ±0.36 ±0.25 ±0.26 ±0.93 ±0.24 ±0.48 ±0.16 ±0.13 ±0.90 ±0.19 ±0.49 ±0.27 ±0.21 ±0.87 ±0.07 ±1.09 ±0.28 ±0.94 ±0.95 ±0.83 ±0.38 ±0.27 ±0.18 ±0.94 ±0.47 ±1.95 ±0.30 ±0.20 ±2.04 ±0.70 ä [ ] p [ ] [ ] [ ] & A 0.3200 20 2900 20 2700 300 30 2400 L 0.8800 20 2800 20 3000 3.5000 30 3400 1.3000 40 1400 40 1900 5.8800 60 5300 400 600 81 81 38 40 4200 5 5 100 4 1200 9 9 120 1900 48 300 10 500 1100 3 ä [ p [ 103 ] ] [ ] [ 5800 ] ↵ > 0.5 ± A ±0.82 ±0.41 ±0.67 ±1.76 L ±0.52 ±0.46 ±0.42 ±1.07 ±0.79 ±0.34 ±0.60 ±1.71 ±0.64 ±0.48 ±0.01 ±2.30 ±1.68 ±1.01 ±1.02 ±2.65 ±2.05
  
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  ] A L ↵ 5 A 2 Mn i, (i = 1, 2,..., n) A i akl akl A, (k, l = 1, 2,..., n) ∆akl ui i A Aui = 0 j i ui. wj A AT ui AT w j = 0 j wj.
  
  wj T u i = ij, wj ij @A @ui @ i ui + A = ui + @ @ @ i @ui. @ wi T wi T @A ui + @ i wi T @ui @ wi T = wi T @ i u i + @ i wi T @ui @, @A @ i ui =. @ @ A akl A akl A @A = [↵ij ] = [ ik jl ], @akl ↵ij = 1 akl (i, j) (k, l) ↵ij = 0 A @ i @A = wi T ui = wik uil /i, k, l 2 {1, 2,..., n}, @akl @akl wik k th lth uil wi akl ui A Γi Γi =  @ i = wi ui T /i, k, l 2 {1, 2,..., n}. @akl Γi akl i akl i ̃i = i ̃i = akl i+ ãkl = akl + ∆akl i N X @ i ∆akl = i+ @akl i+ k,l=1 N X wik uil ∆akl k,l=1 ekl (akl A G alk ) A A a12 ⇠ a21 0 0 B B B B a21 + ⇠ Ã = [ãij ] = B B B B @ an1 a12 + ⇠ ··· ··· ··· 1 a1n C C C C C. C C C A 0 A 0 B ( B B B D̃ = [d̃ij ] = B B B B @ Pn j=1 a1j ) + ⇠ 0 0 ( Ã 0 j=1 a2j ) + ⇠ ··· 0 ··· Pn D̃ ··· Pn j=1 anj 1 C C C C C. C C C A (L̃ = D̃ Ã) 0 B (L11 + ⇠) B B B (L21 + ⇠) L̃ = B B B B @ Ln1 (L12 + ⇠) ··· (L22 + ⇠) ··· ··· 1 L1n C C C C C. C C C A Lnn T race(L̃) Pn ̃ 2 [0, 1] ρ̃ i=1 ρ̃ = ̃i = 1 L̃. T race(L̃) ρ̃ n X @ i ∆Lkl @Lkl k,l=1  n X 1 @ i Pn + = ∆Lkl, i @Lkl T race(L) + k=1 ∆Lkk k,l=1 ̃i = Lkl i 1 T race(L̃)  i+ L L̃ ̃i = 1 T race(L̃)  i+ @ i @ i @ i @ i ∆L11 + ∆L22 + ∆L12 + ∆L21. @L11 @L22 @L12 @L21 ∆L11 = ∆L12 = ∆L22 = ∆L21 = ⇠ ̃i = 1 T race(L̃)  i+ ✓ ◆ @ i @ i @ i @ i + + + ⇠ @L11 @L12 @L22 @L21 ρ̃ S(ρ̃) = n X ̃ i log ̃ i, 2 i=1 S(ρ̃) = n  X i=1 i + (wi1 ui2 + wi2 ui1 + wi1 ui1 + wi2 ui2 )⇠ T race(L) + 2⇠ ⇥log2  i + (wi1 ui2 + wi2 ui1 + wi1 ui1 + wi2 ui2 )⇠ T race(L) + 2⇠. e12 ∆i,j = S(ρ0 ) S(ρ̃[⇠i,j ]), ρ̃[⇠i,j ] ρ0 ai,j, aj,i 2 {0, 1}/ei,j 2 E ⇠i,j 2 [0, 1] ei,j 2 E ⌘i,j = exp (∆i,j ). ⌘ (⇠ = 1) ⌘ 1 ⌘ G = (V, E) V, E A D W L=D A L ρ0 = T race(L) S(ρ0 ) = T race(ρ0 Log2 ρ0 ) ⇠ 2 [0, 1] Ã = [ãij ] = L̃ = D̃ Ã 8 > > <a > > :0 L̃ ρ̃ = T race( L̃) S(ρ̃) = ekl = (vk, vl ) 2 E i, j 2 {k, l} ⇠, ij n P d ̃i = ãij j=1 T race(ρ̃ Log2 ρ̃) Wkl = Wlk = exp(S(ρ0 ) 5 5 10 S(ρ̃[⇠])) ekl 2 E e56 = (v5, v6 ) (A) L=D (L) A ) • ( ) • • • A L L A A L ) ( (,, ) ) ) A (L = D L A) ) (A) (L) ) A L ↵ ) A ) ) VPV L ) (DGA) ( ) (EL ) ( ) ( ) (A) (L) ↵ > 0.5 A • (, L, ) • ↵ • (EA ) (E|L| ) • X Y1 (1 { ↵i )} ↵ ∆ =1 i=1 ↵i ∆i • (A) (L) L=D A↵ = ↵ D (1 A ↵)A. A 21 = 12 L ↵ = 0.5 • A | L | L L |L| • V ul(vi ) = X ⌘i,j vi ⇠vj ei,j 2 E ⌘i,j V ul(vi ) = ⌘i,j, n 1 v ⇠v X i n • 1 j vi {vj /vj 2 V vj 6= vi } ⇡ ⇡ ⇡.