14. xi ∈ R (i ∈ {1, …, n}) :
d
D: R × R → R :
d d
h: Rd × Rd → R :
1 2
h(x, x ) = exp − D (x, x ) .
t
(t 0 : )
A = (h(xi, xj)) ij ∈ R n×n
:
Yi ∈ {1, -1} : xi
r
Y∈R n×r
: Yi
T
i
yk : Y k (k )
14
15. 3 Y
‣
‣
‣
n
1 2
min Yi − Yj Aij
Y 2
i,j=1
n×r
s.t. Y ∈ {1, −1} , 1 Y = 0, Y Y = nIr×r .
(1 : 1 )
Y
15
16. r=1 NP-hard
xi (i ∈ {1, …, n}) Aij
G
‣r = 1
Yi = -1
Yi = 1
16
17. Y
Y
n
1 2
min Yi − Yj Aij
Y 2
i,j=1
s.t. Y∈R n×r
, 1 Y = 0, Y Y = nIr×r .
Y 1, -1
17
18. G A D = diag(A1)
L=D-A G
n
n
n
1 2 2
Yi − Yj Aij = Yi Dii − Yi Yj Aij
2
i,j=1 i=1 i,j=1
= tr(Y DY) − tr(Y AY)
= tr(Y LY).
18
19. L
Y
min tr(Y LY)
Y
s.t. Y ∈ Rn×r , 1 Y = 0, Y Y = nIr×r .
‣ Y k yk L
0 k
- 0 1
‣
19
22. ‣
‣ m (≪ n) u1, …, um ∈ R d
:
‣
‣ s (
s = 2) (2 ) 22
23. Truncated similarity
‣ :
n×m
Z∈R
h(xi ,uj )
h(xi ,uj ) , ∀j ∈ i
Zij = j ∈i
0, otherwise.
( i xi s ( ≪ m) )
‣Z i xi
s 0
23
24. Anchor Graph
Λ = diag(1 Z) ∈ R
T m×m
:Z
^ = ZΛ−1 Z .
A ≤m
G A ^
A
G
‣ 2 Zij
^
A G( )
24
25. ‣ G
‣
L
‣A
^ ( ) L
^ ^ ^
L = diag(A1) − A = I − A
^
A
- L ^
A (1
)
25
26. ‣ M = Λ Z ZΛ ∈ R
-1/2 T -1/2 m×m
‣A^ = ZΛ-1/2Λ-1/2ZT
‣ ZΛ = UΣ V :
-1/2 1/2 T
( U∈R n×m
Σ∈Rm×m
V∈R m×m
)
‣
^ = UΣ1/2 V VΣ1/2 U = UΣU ,
A
M = VΣ1/2 U UΣ1/2 V = VΣV .
‣ U = ZΛ -1/2
VΣ -1/2
‣U r Y 26
27. ‣Σ 1, σ1, …, σr, …
σ1, …, σr V v1, …, vr ∈ R m
‣ Σr = diag(σ1, …, σr) Vr = [v1, …, vr]
‣ W
√ −1/2 −1/2
W = nΛ Vr Σr ∈R m×r
‣ Y
Y = ZW.
27
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