Full uk opportunity presentation 25 feb2013 cuts cuts cuts locked (1)
Bilipschitz embeddings, metric geometry, and sparse cuts in graphs
Transcript of Bilipschitz embeddings, metric geometry, and sparse cuts in graphs
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]