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spectral clustering between friends

spectral clustering between friends. One of these things is not like the other…. spectral clustering (a la Ng-Jordan-Weiss). data. similarity graph. edges have weights w ( i , j ). e.g. the Laplacian. diagonal matrix D. Normalized Laplacian :. energy. Normalized Laplacian :.

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spectral clustering between friends

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  1. spectral clustering between friends

  2. One of these things is not like the other…

  3. spectral clustering (a la Ng-Jordan-Weiss) data similarity graph edges have weights w(i,j) e.g.

  4. the Laplacian diagonal matrix D Normalized Laplacian:

  5. energy Normalized Laplacian:

  6. Normalized Laplacian: Compute first k eigenvectors: v1, v2 , …, vk spectral embedding

  7. clustering Run k–means to cluster the points

  8. it’s amazing! it’s mediocre! spectral clustering … what to prove? it’s antiquated Sidi, et. al. 2011 [TelAviv-SFU] Many, many variants… Many opinions

  9. spectral embedding why should spectral clustering work? k perfect clusters

  10. S Expansion: For a subset SµV, define graph expansion E(S) = set of edges with one endpoint in S.

  11. S1 Expansion: For a subset SµV, define S4 graph expansion S3 E(S) = set of edges with one endpoint in S. S2 k-way expansion constant: Theorem [Cheeger70, Alon-Milman85, Sinclair-Jerrum89]: “most important result in spectral graph theory” -- Wikipedia

  12. S1 Higher-order Cheeger Conjecture [Miclo 08]: S4 For every graph G and k2N, we have Miclo’s conjecture S3 S2 for some C(k) depending only on k. [Lee-OveisGharan-Trevisan 2012]: True with This bound for C(k) is tight. Algorithm of Ng-Jordan-Weiss works, changing the last step.

  13. we do random projection the clustering step random space partition Run k–means to cluster the points

  14. S1 Higher-order Cheeger Conjecture [Miclo 08]: S4 For every graph G and k2N, we have Miclo’s conjecture S3 S2 for some C(k) depending only on k. [Lee-OveisGharan-Trevisan 2012]: True with This bound for C(k) is tight. Algorithm of Ng-Jordan-Weiss works, changing the last step.

  15. Suppose the data has some nice low-dimensional structure hybrid algorithms Spectral embedding could lose that information: Back in a high-dimensional space

  16. Suppose the data has some nice low-dimensional structure hybrid algorithms Use spectral embedding distances to deform the data Do clustering on transformed data set

  17. unraveling the mysteries of complexity

  18. Consider linear equations in two variables, modulo a prime p Variables: x1, x2, …, xn the unique games conjecture x12+x2=4 x4–3x7 =1 x9+8x12 =9 … If there exists a solution that satisfies 99% of the equations, can you find one that satisfies 10%? Conjectured to be NP-hard [Khot 2002]

  19. Construct a graph with one vertex for every variable, and an edge whenever two variables occur in the same constraint. a spectral attack x12+x2=4 x4–3x7 =1 x9+8x12 =9 … A “good” solution to the equations implies a partition of the graph into p nice clusters!

  20. S1 Higher-order Cheeger Theorem: S4 For every graph G and k2N, we have a spectral attack S3 S2 Unnecessary for large k: [Arora-Barak-Steurer 2010] A better asymptotic dependence would disprove the UGC.

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