multi-way spectral partitioning

and higher-order Cheeger inequalities

University of Washington

James R. Lee

Stanford University Luca Trevisan

Shayan Oveis Gharan

Si

spectral theory of the Laplacian

Consider a d-regular graph G = (V, E).

kf(x)¡ f(y)k1 =

Define the normalized Laplacian:

(where A is the adjacency matrix of G)

L is positive semi-definite with spectrum

Fact: is disconnected.

¸2 = minf?1

hf; Lfihf; fi = min

f?1

1

d

X

u»vjf(u)¡ f(v)j2

X

u2Vf(u)2

Cheeger’s inequality

Expansion: For a subset S µ V, define

E(S) = set of edges with one endpoint in S.

S

Theorem [Cheeger70, Alon-Milman85, Sinclair-Jerrum89]:

¸2

2· ½G(2) ·

p2¸2

½G(k) = minfmaxÁ(Si) : S1; S2; : : : ; Sk µ V disjointgk-way expansion constant:

S2

S3

S4

is disconnected.

Miclo’s conjecture

has at least k connected components.

f1, f2, ..., fk : V R first k (orthonormal) eigenfunctions

spectral embedding: F : V Rk

Higher-order Cheeger Conjecture [Miclo 08]:

¸k

2· ½G(k) · C(k)

p¸k

for some C(k) depending only on k.

For every graph G and k 2 N, we have

our results

¸k

2· ½G(k) · O(k2)

p¸k

Theorem: For every graph G and k 2 N, we have

½G(k) = minfmaxÁ(Si) : S1; S2; : : : ; Sk µ V disjointg

½G(k) · O(p

¸2k logk)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

½G(k) · O(p

¸2k)

small set expansion

Á(S) · O(p

¸k logk)

Corollary: For every graph G and k 2 N, there is a subset

of vertices S such that and,

Previous bounds:

jSj ·npk

Á(S) · O(p

¸k logk)and

[Louis-Raghavendra-Tetali-Vempala 11]

jSj ·n

k0:01 Á(S) · O(p

¸k logk n)and

[Arora-Barak-Steurer 10]

proof

½G(k) · O(k2)p

¸k

Theorem: For every graph G and k 2 N, we have

proof

½G(k) · kO(1)p

¸k

Theorem: For every graph G and k 2 N, we have

Dirichlet Cheeger inequality

For a mapping , define the Rayleigh quotient:

R(F ) =

1

d

X

u»vkF (u)¡ F (v)k2

X

u2VkF (u)k2

Lemma: For any mapping , there exists a subset

such that:

Miclo’s disjoint support conjecture

Conjecture [Miclo 08]:

For every graph G and k 2 N, there exist disjointly supported

functions so that for i=1, 2, ..., k, Ã1; Ã2; : : : ;Ãk : V !R

Localizing eigenfunctions:

Rk

isotropy and spreading

Isotropy: For every unit vector x 2 Rk

M =

0BBB@

f1f2...

fk

1CCCA

Total mass:

radial projection

Define the radial projection distance on V by,

dF (u; v) =

°°°°F (u)

kF (u)k ¡F (v)

kF (v)k

°°°°

Fact:

Isotropy gives: For every subset S µ V,

diam(S; dF ) ·1

2=)

X

v2SkF (v)k2 ·

2

k

X

v2VkF (v)k2

smooth localization

Want to find k regions S1, S2, ..., Sk µ V such that,

mass:

separation:

Then define by, Ãi : V ! Rk

so that are disjointly supported and

Si

"(k)=2 Sj

smooth localization

Want to find k regions S1, S2, ..., Sk µ V such that,

mass:

Claim:

Then define by, Ãi : V ! Rk

so that are disjointly supported and

separation:

disjoint regions

Want to find k regions S1, S2, ..., Sk µ V such that,

mass:

separation:

For every subset S µ V,

diam(S; dF ) ·1

2=)

X

v2SkF (v)k2 ·

2

k

X

v2VkF (v)k2

How to break into subsets? Randomly...

random partitioning

Rk

random partitioning

Rk spreading )

separation

smooth localization

We found k regions S1, S2, ..., Sk µ V such that,

mass:

separation:

Si Sj

improved quantitative bounds

- Take only best k/2 regions (gains a factor of k)

½G(k) · O(p

¸2k logk)

- Before partitioning, take a random projection into O(log k) dimensions

Recall the spreading property: For every subset S µ V,

diam(S; dF ) ·1

2=)

X

v2SkF (v)k2 ·

2

k

X

v2VkF (v)k2

- With respect to a random ball in d dimensions...

u v

u

v

k-way spectral partitioning algorithm

ALGORITHM:

1) Compute the spectral embedding

2) Partition the vertices according to the radial projection

3) Peform a “Cheeger sweep” on each piece of the partition

planar graphs: spectral + intrinsic geometry

2) Partition the vertices according to the radial projection

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]

planar graphs: spectral + intrinsic geometry

2) Partition the vertices according to the radial projection

[Jordan-Ng-Weiss 01]

open questions

1) Can this be made ?

2) Can [Arora-Barak-Steurer] be done geometrically?

We use k eigenvectors, find ³ k sets, lose .

What about using eigenvectors to find sets?

3) Small set expansion problem

There is a subset S µ V with

and

Tight for (noisy hypercubes)

Tight for (short code graph)

[Barak-Gopalan-Hastad-Meka-Raghvendra-Steurer 11]