1.
Hashing with Graphs
Anchor Graph Hashing
ICML2011 2011 8 4
blog.beam2d.net @beam2d

2.
Hashing with Graphs.
Liu, W., Wang, J., Kumar, S. and Chang, S.-F. ICML 2011.
2

3.
Hashing
d
R
‣
( , )
‣
3

4.
Hashing
d
R
‣ x∈R
d
y ∈ {1, -1}
r
( 0,1 ±1 )
‣
- 4

5.
Hashing
d
R
‣ ( etc.)
‣r
‣ , r
5

6.
Hashing = ○
‣
‣
-
-
( )
- →
-
‣ :
- n
- n 6

7.
(xi)i=1,...,n ,
Y = (Yik)i=1,...,n, k=1,...,r
‣ (1)
‣ (2) 2
Aij = exp(− xi − xj /t)
‣ (3)
n ( , )
1 2
min Yi − Yj Aij Yi: Y i
Y 2
i,j=1
(1)
n×r
s.t. Y ∈ R , 1 Y = 0, Y Y = nIr×r .
(2) (3)
(1 : 1 )
7

8.
G A , D = diag(A1)
,L=D-A G
, A
.
n
1 2
Y i − Yj Aij = tr(Y LY).
2
i,j=1
8

9.
L
min tr(Y LY)
Y
(2) (3)
s.t. Y ∈ Rn×r , 1 Y = 0, Y Y = nIr×r .
‣ (3) ,L
r (cf. )
‣ 1 (2)
‣2 1 , (2)
‣2
9

10.
‣A L
‣ A
‣ m (≪ n) u1, …, um
d
∈R
‣
-
- n=69,000
m=300
‣ s
( : s=2) 10

11.
‣2
‣
-
n×m
Z∈R
-
-Z ( s/m)
h(xi ,uj )
h(xi ,uj ) , ∀j ∈ i
Zij = j ∈ i
0, otherwise.
( i : xi s , h: ) 11

12.
‣2
,
‣ Λ = diag(1 Z) ∈ R
T m×m
,
^ = ZΛ−1 Z .
A
‣
12

13.
‣ ^
A
- ( 0)
- m
-( ) . ^
L=I- A
,L ^
A
- ( )
^
‣ A
- ,
13

14.
‣ ^ = ZΛ-1/2Λ-1/2ZT
A
‣ M = Λ Z ZΛ
-1/2 T -1/2
∈R
m×m
‣ ZΛ -1/2
= UΣ
1/2 T
V :
n×m m×m m×m
( U∈R Σ∈R V∈R )
‣
^ = UΣ1/2 V VΣ1/2 U = UΣU ,
A
M = VΣ1/2 U UΣ1/2 V = VΣV .
‣ U = ZΛ
-1/2
VΣ
-1/2
14
‣U r Y

15.
‣Σ 1, σ1, …, σr, …
m
σ1, …, σr V v1, …, vr ∈ R
‣ Σr = diag(σ1, …, σr) Vr = [v1, …, vr]
‣ W
√ −1/2 −1/2 m×r
W = nΛ Vr Σr ∈R
‣ Y
Y = ZW.
15

16.
Nyström method
‣ ,
‣ n→∞
-
-
‣ ,n k
φn, n
k
1 ^
φn,k (x) = A(x, xi )Yik .
σk
i=1
^
A ():
16

17.
AGH Nyström
n
1 ^
φn,k (x) = A(x, xi )Yik .
σk
i=1
‣
‣
φn,k (x) = wk z(x). z: x
-
- O(dm)
17

18.
‣ : *( )
‣ 28 × 28 = 784
‣ (784 )
‣ n = 69,000
‣ 1,000
‣
* http://yann.lecun.com/exdb/mnist/ 18

19.
( )
m = 300, s = 2 19

20.
‣
‣
‣
‣
‣
‣
20

21.
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approximate nearest neighbor in high dimensions. Proceedings of
FOCS, 2006.
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P. Indyk and R. Motwani. Approximate nearest neighbor: Towards
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embeddings. NIPS 22, 2010.
B. Kulis and K. Grauman. Kernelized locality-sensitive hashing for
scalable image search. Proceedings of ICCV, 2009. 21

22.
W. Liu, J. He, and S.-F. Chang. Large graph construction for scalable
semi-supervised learning. Proceedings of ICML, 2010.
W. Liu, J. Wang, S. Kumar, and S.-F. Chang. Hashing with graphs.
ICML, 2011.
M. Raginsky and S. Lazebnik. Locality-sensitive binary codes from
shift-invariant kernels. NIPS 22, 2010.
J. Wang, S. Kumar, and S.-F. Chang. Sequential projection learning for
hashing with compact codes. Proceedings of ICML, 2010.
Y. Weiss, A. Torralba, and R. Fergus. Spectral hashing. NIPS 21, 2009.
C. Williams and M. Seeger. The effect of the input density distribution
on kernel-based classifiers. Proceedings of ICML, 2000.
22