1. spectral clustering between friends

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

5. the Laplacian diagonal matrix D

6. energy

7. spectral embedding Compute first k eigenvectors: v 1, v 2, …, v k

8. clustering Run k –means to cluster the points

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

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

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

12. graph expansion Expansion: For a subset S µ V, define E ( S ) = set of edges with one endpoint in S. S1S1 Theorem [Cheeger70, Alon-Milman85, Sinclair-Jerrum89] : k-way expansion constant: S2S2 S3S3 S4S4 “most important result in spectral graph theory” -- Wikipedia

13. Miclo’s conjecture Higher-order Cheeger Conjecture [Miclo 08]: for some C ( k ) depending only on k. For every graph G and k 2 N, we have [Lee-OveisGharan-Trevisan 2012]: True with This bound for C ( k ) is tight. Algorithm of Ng-Jordan-Weiss works, changing the last step. S1S1 S2S2 S3S3 S4S4

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

15. Miclo’s conjecture Higher-order Cheeger Conjecture [Miclo 08]: for some C ( k ) depending only on k. For every graph G and k 2 N, we have [Lee-OveisGharan-Trevisan 2012]: True with This bound for C ( k ) is tight. Algorithm of Ng-Jordan-Weiss works, changing the last step. S1S1 S2S2 S3S3 S4S4

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

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

18. unraveling the mysteries of complexity

19. the unique games conjecture Consider linear equations in two variables, modulo a prime p Variables: x 1, x 2, …, x n x 12 + x 2 = 4 x 4 – 3 x 7 = 1 x 9 + 8 x 12 = 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]

20. a spectral attack Construct a graph with one vertex for every variable, and an edge whenever two variables occur in the same constraint. x 12 + x 2 = 4 x 4 – 3 x 7 = 1 x 9 + 8 x 12 = 9 … A “good” solution to the equations implies a partition of the graph into p nice clusters!

21. a spectral attack Higher-order Cheeger Theorem : For every graph G and k 2 N, we have S1S1 S2S2 S3S3 S4S4 Unnecessary for large k : [Arora-Barak-Steurer 2010] A better asymptotic dependence would disprove the UGC.