multi-way spectral partitioning and higher-order Cheeger inequalities

1. James R. Lee University of Washington multi-way spectral partitioningand higher-order Cheeger inequalities Shayan Oveis Gharan Luca Trevisan Stanford University Si

2. Consider a d-regular graph G = (V, E). Define the normalized Laplacian: spectral theory of the Laplacian (where A is the adjacency matrix of G) L is positive semi-definite with spectrum is disconnected. Fact:

3. S Expansion: For a subset SµV, define S4 Cheeger’s inequality S3 E(S) = set of edges with one endpoint in S. S2 k-way expansion constant: is disconnected. Theorem [Cheeger70, Alon-Milman85, Sinclair-Jerrum89]:

4. has at least k connected components. f1, f2, ..., fk : VRfirst k (orthonormal) eigenfunctions Miclo’s conjecture spectral embedding: F : VRk Higher-order Cheeger Conjecture [Miclo 08]: For every graph G and k2N, we have for some C(k) depending only on k.

5. Theorem: For every graph G and k2N, we have our results Also, - actually, can put for any ±>0 - tight up to this (1+±) factor - proved independently by [Louis-Raghavendra-Tetali-Vempala 11] If G is planar (or more generally, excludes a fixed minor), then

6. Corollary: For every graph G and k2N, there is a subset of vertices S such that and, small set expansion Previous bounds: and [Louis-Raghavendra-Tetali-Vempala 11] and [Arora-Barak-Steurer 10]

7. Theorem: For every graph G and k2N, we have proof

8. Theorem: For every graph G and k2N, we have proof

9. For a mapping , define the Rayleigh quotient: Dirichlet Cheeger inequality Lemma: For any mapping , there exists a subset such that:

10. Conjecture [Miclo 08]: For every graph G and k2N, there exist disjointly supported functions so that for i=1, 2, ..., k, Miclo’s disjoint support conjecture Localizing eigenfunctions:

11. Isotropy: For every unit vector x 2Rk isotropy and spreading Total mass:

12. Define the radial projection distance on V by, radial projection Fact: Isotropy gives: For every subset SµV,

13. Want to find k regions S1, S2, ..., SkµVsuch that, mass: smooth localization separation: Then define by, so that are disjointly supported and Sj Si

14. Want to find k regions S1, S2, ..., SkµVsuch that, mass: smooth localization separation: Then define by, so that are disjointly supported and Claim:

15. Want to find k regions S1, S2, ..., SkµVsuch that, mass: disjoint regions separation: For every subset SµV, How to break into subsets? Randomly...

16. random partitioning

17. spreading ) random partitioning separation

18. We found k regions S1, S2, ..., SkµVsuch that, mass: smooth localization separation: Sj Si

19. - Take only best k/2 regions (gains a factor of k) improved quantitative bounds - Before partitioning, take a random projection into O(log k) dimensions Recall the spreading property: For every subset SµV, - With respect to a random ball in d dimensions... v v u u

20. ALGORITHM: 1) Compute the spectral embedding k-way spectral partitioning algorithm 2) Partition the vertices according to the radial projection 3) Peform a “Cheeger sweep” on each piece of the partition

21. 2) Partition the vertices according to the radial projection planar graphs: spectral + intrinsic geometry For planar graphs, we consider the induced shortest-path metric on G, where an edge {u,v} has length dF(u,v). Now we can analyze the shortest-path geometry using [Klein-Plotkin-Rao 93]

22. 2) Partition the vertices according to the radial projection planar graphs: spectral + intrinsic geometry [Jordan-Ng-Weiss 01]

23. 1) Can this be made ? 2) Can [Arora-Barak-Steurer] be done geometrically? open questions We use k eigenvectors, find ³k sets, lose . 3) Small set expansion problem What about using eigenvectors to find sets? There is a subset SµVwith and Tight for (noisy hypercubes) Tight for (short code graph) [Barak-Gopalan-Hastad-Meka-Raghvendra-Steurer 11]