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1. 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. 2. b-bit Minwise Hashing for Estimating Three-Way Similarities • Minwise Hashing (MinHash) ? • b-bit Minwise Hasing ?2011 2 14
3. 3. Motivation • • , • • Web • • • 2 ( ) • Minwise Hasing (MinHash) [Broder 1997] sign random projections (simhash) Hamming Distance LSH2011 2 14
4. 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 • Jaccard2011 2 14
5. 5. • • Hash bit • Altavista 40bit Fetterly WWW03 64bit • Hash 1 or 2bit •2011 2 14
6. 6. • 2 • • Jaccard (0.5 )2011 2 14
7. 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. 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 funmical data Registration basedtional geometry geometry matches the diffusion maps of fMRI on the functional alignment Integrating functional features into the registration process prombrain 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-sumical2 registration and the proposed functional geometry alignment. Func- warp to 2011 14 points, while at the same time regularizing the surface
9. 9. Motivation • • fMRI • • ? Above-threshold region in Above-threshold region in2011 2 14 source subject target subject
10. 10. • fMRI • Voxel ( Kernel) • Diffusion Maps • Voxel a. Maps of two subjects s0 Ψ0 Ψ1 s1 Subject 1 Map 1 Map 2 Subject 22011 2 14 b. Aligning the point sets
11. 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. 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 0.12011 2 14 source subject target subject
13. 13. 2011 2 14