1 ! " !" #$ % & $ ' (!$ )!$ *!$ +!" & & " ,! -$ A ' m × n & r ≤ min{m, n} % . r . 2r r / & r 0 " A$ & (m + n)r 1 2 $ 0 Ck ' (m + n)k$ k < r$ 3% $ . 0 4 . !$ !" 5& 0 0 Ck A$ 0 4 Ck $ 0 ' 0 A % $ 0 $ % . 6 4 6$ 3 0 . 0 6$ " 0 5$ 78$ $ " 9!" W(Ck ) . 6 CK $ Ck / 3 W(Ck ) 2 . .$ 3$ " 4 $ % .$ / k σi A Ck Ck . 6 - 3$ Ck & k 3 k $ ' m × n & r ≤ k 4$ σi A Ak 7 % $ 512 × 512 3 : ;! A$ ' m × n$ < . = A = USV T " U ' m × m$ V ' n × n & S m × n$ $ $ 0 σi $ i = 1, 2, · · · , min{m, n}$ 0 A σi = σ1 ≥ σ2 ≥ . . . 4 % {σi } A ui U vi V " 0 /" A - i$ (ui , σi , vi ) i A 4 > A ? $ 0 A r = rango(A) $ .= A= r σi ui viT . (" i=1 (ui , σi , vi ) i A 2" $ A 7 σi ui viT ;!= Ak = k σi ui viT , k < r. 9" i=1 0 Ak k 4 < A - Ak < A ! A = (aij ) & B = (bij ) ' m × n$ > = m,n i,j=1 (aij − bij ) m,n 2 i,j=1 aij D(A, B) = 2 ;" 7 $ D(A, B) ≤ 0.05$ B < " A 4 . ! 0 Ak < A$ = r σi2 i=k+1 r 2 i=1 σi D(A, Ak ) = ," - $ D(A, Ak ) 0 A ' k(m + n)= Ck A ∈ Rm×n $ r$ k < r 3 uT1 v1T uT vT 2 2 Ck = uTk vkT )" (u1 , . . . , uk ) k U & (v1 , . . . , vk ) k V m & n > 0 1 ≤ m < n 4= m (m + n) < m n " 2 = 1 1 ≤ m < n$ < > > p 1 1 > 2n & 0 m + p = n$ m + n = n − p + n = 2n − p < 2n$ m+n mn 0 mn m mn > = m+n 2n 2 *" A ' m × n 3 ≤ m ≤ n r ≥ m/2 < Ck )" 0 ' 0 A = @ m −1 m 2 k= +" m−1 m 2 0 = k< m 2 " & r ≥ m/2 k < r & > 3 2$ ' Ck k(m + n) A m < n$ 0 " & 9$ = m (m + n)k < (m + n) < mn " 2 0 / m = n 4 ' Ck / k(m + n) = k2n < n2 (2n) = n2 $ 0 ' A$ 0 4 ( < $ $ Ck r ≥ m/2 1/ $ & & Ck ' 0 A 4 $ = / ( & k ≥ 1$ +"$ < i = 1, . . . , k Ci ' 0 A B 0 A ' 0 m > n$ ( AT - $ 0 9 3 9 " A ( r ≥ m/2 4 3 $ $ $ " 0 3 ! " # C ' 512 × 512 512 498 × 621 498 256 × 256 239 359 × 657 359 $ $% & A 0 3 / ( W(A) . A$ ' TA $ ' 0 A$ & 3 % 3 W(A) 0 A = W −1 (TA ) < A 7 A A & TA 0 0 . " 4 $ % . W = A ( 3 ( & ($ < Ck ' 0 A - & ," & ( 3 k < r 9 . W Ck $ W(Ck ) TCk . 3 W(Ck ) 3 TCk 0 % k σi ; - $ . k 3 k &$ 0 C W −1 (TCk ) = C ($ 9" AK 0 < A 4 % / W A & Ck 1 / TA & TCK $ CK ' 0 A$ @$ . W(Ck ) Ck $ & k 0 Ak 3 A$ . < . 6 ( 3 . " 6 & ! - 3% . 6 ( D . A ' m × n = A −→ W(A) = h1 d1 a1 v 1 (" h1 $ d1 $ a1 $ & v 1 ' m/2×n/2 & 8 /$ 8 $ & 8 $ a1 $ & '$ αi $ i = 1, . . . , m × n W(A)$ 0 = A= m×n αi Wi = H 1 + D 1 + A1 + V 1 9" i=1 Wi $ 0 . 6 & $ . Rm×n A1 < A 7 < $ = A1 = W −1 O O a1 O ≡ W −1 (TA ) ≡ A ;" . 6$ > 3 ' a1 $ m4n 1$ TA 3 a1 $v 1 & h1 $ > 3 A A1 $ V 1 & H 1 3m4 n / W a1 & . 6 ($ > 3 5 A = (aij ) 0 4 512 × 512$ & > 0 $ $ aij ∈ [0, 255] A r = 507 &$ $ ( 5 k = 80 & 0 A80 0 D(A, A80 ) = 0.04$ 0 A80 <$ $ A ! " 4 ( / . 6 ($ > ?1@1E & ( 4 ( TA & TCk % ' $ 0 ' A & Ck $ 0 = mn/(k(m + n)) = 512 ∗ 512/(80 ∗ 2 ∗ 512) = 512/160 ≈ 3 - $ < % 3 0 W(A) @ /2 0 0 /& 0 k $ 3 7 % < $ ( $ > 0 . ;"$ = D(A, A) & . 6 AF D(A, Ck ) 0 C k ) 0 & $ & D(A, . 6 A 7 ? B> 3 TA 1 1 1 a$v &h 3m n/4 ( a1 & h1 m n/2 1 9 a m n/4 ; a2 $ v 1 & h1 9m n/16 , a2 $ h2 $ v 1 & h1 5m n/8 D(A, A) 0.013 0.035 0.043 0.070 0.058 k ) D(A, C 0.048 0.052 0.82 0.25 0.069 $ # ' k = 80 C k ) D(A, 0.047 0.042 0.82 0.25 0.059 # ( A )*"+ & C80 , - ( A )*"+ & C80 . 4 $ ( & , 3 (" 0 % 0 % 9 & ; 3 9" 0 3 W(A) 2 $ 0 3 W(Ck ) 4 $ A CK ; & , //F / 8 & W(Ck )$ 0 / W(A) 4 % 0 . W(Ck ) : 0 D(A, Ck ) C k )$ $ $ 0 D(A, / DV S 6$ 3 Ck 0 / 3 0 Ck ' 0 $ $ 0 3 1$ $ $ / 0 0 . 6 A Ck 7 . k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