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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]
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...
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]