1. NIPS-2010
@
• b-bit Minwise Hashing for Estimating Three-
Way Similarities. P. Li et al.
•
• Functional Geometry Alignment and
Localization of Brain Areas. Langs et al.
•
2011 2 14

2. b-bit Minwise Hashing for
Estimating Three-Way
Similarities
• Minwise Hashing (MinHash) ?
• b-bit Minwise Hasing ?
2011 2 14

3. Motivation
•
• ,
•
• Web
•
•
• 2 ( )
• Minwise Hasing (MinHash) [Broder 1997]
sign random projections (simhash) Hamming
Distance LSH
2011 2 14

4. Minwise Hashing
• Jaccard
|A ∩ B|
J(A, B) =
|A ∪ B|
•
• Random parmutation (or Hash ) π(x)
• A π(x) Pr[min(π(A)) = min(π(B))]
Pr[min(π(A)) = min(π(B))] = J(A, B)
• A = {1, 3, 5, 7}, B = {3, 4, 5}
⇒ A ∩ B = {3, 5}, A ∪ B = {1, 3, 4, 5, 7}
• min(h(A)) = min(h(B)) {1,3,4,5,7}
3 5
• Jaccard
2011 2 14

5. •
•
Hash bit
• Altavista 40bit Fetterly
WWW03 64bit
• Hash 1 or 2bit
•
2011 2 14

6. • 2
•
• Jaccard (0.5 )
2011 2 14

7. • b-Bit Minwise Hashing for Estimating
Three-Way Similarities NIPS2010
• b-Bit Minwise Hashing 3
Jaccard
|A ∩ B ∩ C|
J(A, B, C) =
|A ∪ B ∪ C|
•
2011 2 14

8. Functional Geometry Alignment
and Localization of Brain Areas
Registration based on anatomical data Registration based on the function
brain 1 registration brain 2 brain 1 embedding re
Figure 1: Standard anatomical registration and the proposed fun
mical data Registration basedtional geometry geometry matches the diffusion maps of fMRI
on the functional alignment
Integrating functional features into the registration process prom
brain 2 brain 1 embedding proposed methods match the centers of activated cortica
cently registration embedding brain 2
correspondences of cortical surfaces [18]. The fMRI signals at t
vector, and registration is performed by maximizing the inter-su
mical2 registration and the proposed functional geometry alignment. Func- warp to
2011 14
points, while at the same time regularizing the surface

9. Motivation
•
• fMRI
•
•
?
Above-threshold region in Above-threshold region in
2011 2 14 source subject target subject

10. • fMRI
• Voxel ( Kernel)
• Diffusion Maps
• Voxel
a. Maps of two subjects
s0 Ψ0 Ψ1 s1
Subject 1 Map 1 Map 2 Subject 2
2011 2 14 b. Aligning the point sets

11. Diffusion Maps
• Coifman and Lafon. Applied and Comp. Harmonic Analysis. 2006
• PCA Isomap
• Spectral Clustering
•
• i,j t i Markov
chain random walk t
• Normalized Graph Laplacian
•
Diffusion Distance
• Diffusion Distance
N (N )
2011 2 14

12. a. Maps of two subjects
s0 Ψ0 Ψ1 s1
Subject 1 Map 1 Map 2 Subject 2
b. Aligning the point sets
xk
0
xl
1
A.
FGA
0.2
0.2
?
Figure 2: Maps of two subjects in the process of registration: (a) Left and right: the0.15 axial and
0.15
sagittal views of the points in the two brains. The two central columns show plots of the first
three dimensions of the embedding in the functional geometry after coarse rotational alignment. (b)
During alignment, a maps is represented as a Gaussian mixture model. The colors in both plots
indicate clusters which are only region in visualization. Above-threshold region in
Above-threshold used for 0.1
0.1
2011 2 14 source subject target subject

13. 2011 2 14