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## Theoretical Computer Science methods in asymptotic geometry

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### Cayley expanders in non-simple groups. Belief Propagation in Codes ... Torus. Surface blocking all. cycles that wrap around. Probabilistic construction of spine ... – PowerPoint PPT presentation

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Title: Theoretical Computer Science methods in asymptotic geometry

1
Theoretical Computer Science methods in
asymptotic geometry
• Avi Wigderson
• IAS, Princeton
• For Vitali Milmans 70th birthday

2
Three topics Methods and Applications
• Parallel Repetition of games and
• Periodic foams
• Zig-zag Graph Product and
• Cayley expanders in non-simple groups
• Belief Propagation in Codes and
• L2 sections of L1

3
Parallel Repetition of Games and Periodic
Foams
4
Isoperimetric problem Minimize surface
area given volume.
One bubble. Best solution Sphere
5
Many bubbles Isoperimetric problem
Minimize surface area given volume.
Why? Physics, Chemistry,
Engineering, Math Best solution? Consider R3
Kelvin 1873 Optimal
Wearie-Phelan 1994 Even better
6
Our Problem
Minimum surface area of body tiling Rd with
period Zd ? d2 area
4
gt4
Choe89 Optimal!
7
Bounds in d dimensions
OPT
OPT
Spherical Cubes exist! Probabilistic
construction! (simpler analysis
Alon-Klartag) OPEN Explicit?
8
Randomized Rounding
Round points in Rd to points in Zd such that for
every x,y 1. 2.
x
y
1
9
Spine
Surface blocking all cycles that wrap around
Torus
10
Probabilistic construction of spine
Step 1 Probabilistically construct B, which in
expectation satisfies
B
Step 2 Sample independent translations of B
until 0,1)d is covered, adding new boundaries
to spine.
11
Linear equations over GF(2)
m linear equations Az b in n variables
z1,z2,,zn Given (A,b) 1) Does there exist z
satisfying all m equations? Easy Gaussian
elimination 2) Does there exist z satisfying
.9m equations? NP-hard PCP Theorem
AS,ALMSS 3) Does there exist z satisfying .5m
equations? Easy YES! Hastad ??gt0, it
is NP-hard to distinguish (A,b) which are not
(½?)-satisfiable, from those (1-?)-satisfiable!

12
Linear equations as Games
Game G Draw j ? m at random Xij
Yij Alice
Bob aj
ßj Check if aj ßj bj Pr YES 1-?
2n variables X1,X2,,Xn, Y1,Y2,,Yn m linear
equations Xi1 Yi1 b1 Xi2 Yi2 b2 .. Xim
Yim bm Promise no setting of the Xi,Yi
satisfy more than (1-?)m of all equations
13
Hardness amplification by parallel repetition
Game Gk Draw j1,j2,jk ? m at random Xij1Xij2
Xijk Yij1Yij2 Yijk
Alice Bob aj1aj2 ajk
ßj1ßj2 ßjk Check if ajt ßjt bjt
?t? k PrYES (1-?2)k Raz,Holenstein,Rao
PrYES (1-?2)k
2n variables X1,X2,,Xn, Y1,Y2,,Yn m linear
equations Xi1 Yi1 b1 Xi2 Yi2 b2 .. Xim
Yim bm Promise no setting of the Xi,Yi
satisfy more than (1-?)m of all equations
Feige-Kindler-ODonnell
Spherical Cubes ?
Raz
X
KORWSpherical Cubes ?
14
Zig-zag Graph Product and Cayley expanders in
non-simple groups
15
Expanding Graphs - Properties
• Geometric high isoperimetry
• Probabilistic rapid convergence of random walk
• Algebraic small second eigenvalue ? 1

Theorem. Cheeger, Buser, Tanner, Alon-Milman,
Alon, Jerrum-Sinclair, All properties are
equivalent!
Numerous applications in CS Math! Challenge
Explicit, low degree expanders H n,d, ?-graph
n vertices, degree d, ?(H)? ?lt1
16
Algebraic explicit constructions Margulis
73,Gaber-Galil,Alon-Milman,Lubotzky-Philips-Sarna
k,Nikolov,Kassabov,,Bourgain-Gamburd 09,
Many such constructions are Cayley graphs.
G a finite group, S a set of generators.
Def. Cay(G,S) has vertices G and edges (g, gs)
for all g ? G, s ? S?S-1.
Theorem. LPS Cay(G,S) is an expander family.
17
Algebraic Constructions (cont.)
• Margulis SLn(p) is expanding (n3 fixed!), via
property (T)
• Lubotzky-Philips-Sarnak, Margulis SL2(p) is
expanding
• Kassabov-Nikolov SLn(q) is expanding (q fixed!)
• Kassabov Symmetric group Sn is expanding.
•
• Lubotzky All finite non-Abelian simple groups
expand.
• Helfgot,Bourgain-Gamburd SL2(p) with most
generators.
• Abelian groups of size n require gtlog n
generators
• k-solvable gps of size n require gtlog(k)n gens
LW
• Some p-groups (eg SL3(pZ)/SL3(pnZ) ) expand with
• O(1) generating sets (again relies on property
T).

18
Explicit Constructions (Combinatorial) -Zigzag
K an n, m, ?-graph. H an m, d, ?-graph.
Edges
Combinatorial construction of expanders.
19
Iterative Construction of Expanders
K an n,m,?-graph. H an m,d,? -graph.
The construction A sequence K1,K2, of expanders
• K1 H2

RVW Ki is a d4i, d2, ½-graph.
20
Semi-direct Product of groups
A, B groups. B acts on A. Semi-direct product
A x B
Connection semi-direct product is a special
case of zigzag Assume ltTgt B, ltSgt A , S sB
(S is a single B-orbit)
Alon-Lubotzky-W Expansion is not a group
property
Meshulam-W,Rozenman-Shalev-W Iterative
construction of Cayley expanders in
non-simple groups.
Construction A sequence of groups G1, G2 , of
groups, with generating sets T1,T2,  such that
Cay(Gn,Tn) are expanders. Challenge Define
Gn1,Tn1 from Gn,Tn
21
Constant degree expansion in iterated
wreath-products Rosenman-Shalev-W
Kassabov Iterate Gn1 SYMd x Gnd Get
(G1 ,T1 ), (G2 ,T2),, (Gn ,Tn ),... Gn
automorphisms of d-regular tree of height n.
Cay(Gn,Tn ) expands ? few expanding orbits for
Gnd
Theorem RSW Cay(Gn, Tn) constant degree
expanders.
22
Near-constant degree expansion in solvable
groups Meshulam-W
Gn x FpGn Get (G1 ,T1 ), (G2 ,T2),, (Gn ,Tn
),... Cay(Gn,Tn ) expands ? few expanding orbits
for FpGn Conjecture (true for Gns)
Cay(G,T) expands ? G has exp(d) irreducible
reps of every dimension d.
Theorem Meshulam-W Cay(Gn,Tn) with
near-constant degree Tn ? O(log(n/2) Gn)
(tight! Lubotzky-Weiss )
23
Belief Propagation in Codes and L2 sections of L1
24
Random Euclidean sections of L1N
• Classical high dimensional geometry
• Kashin 77, Figiel-Lindenstrauss-Milman
77
• For a random subspace X ? RN with dim(X) N/2,
• L2 and L1 norms are equivalent up to universal
factors
• x1 T(vN)x2 ?x?X
• L2 mass of x is spread across many coordinates
• i xi vNx2
O(N)
• Analogy error-correcting codes Subspace C of
F2N with every nonzero c ? C has ?(N) Hamming
weight.

25
Euclidean sections applications
• Low distortion embedding L2 ? L1
• Efficient nearest neighbor search
• Compressed sensing
• Error correction over the Reals.
•
• Challenge Szarek, Milman, Johnson-Schechtman
find an efficient, deterministic section with
L2L1
• X ? RN dim(X) vs. ?istortion(X)
• ?(X) Maxx?
X(vNx2)/x1
• We focus on dim(X)?(N) ?(X) O(1)

26
Derandomization results Arstein-Milman
• For
• dim(X)N/2 ?(X) (vNx2)/x1 O(1)
• X ker(A)

• random bits
• Kashin 77, Garnaev-Gluskin 84
O(N2 )
• A a random sign matrix.
• Arstein-Milman 06
O(N log N)
• Expander walk on As columns
• Lovett-Sodin 07
O(N)
• Expander walk k-wise independence
• Guruswami-Lee-W 08 ?(X) exp(1/?) N?
??gt0
• Expander codes belief propagation

27
• Key ideas Guruswami-Lee-Razborov
• L ? Rd is (t,?)-spread if every x ? L,
• ?S ? d, St xS2 (1-?)x
• No t coordinates take most of the mass
• Equivalent notion to distortion (and easier to
work with)
• O(1) distortion ? ( ?(d), ?(1) )-spread
• (t, ?)-spread ? distortion O(?-2 (d/t)1/2)
• Note Every subspace is trivially (0, 1)-spread.
• Strategy Increase t while not losing too much L2
mass.

28
Constant distortion construction GLW (like
Tanner codes)
Ingredients for XX(H,L) - H(V,E) a d-regular
expander - L ? Rd a random subspace X(H,L)
x?RE xE(v) ? L ?v? V Note - N E
nd/2 - If L has O(1) distortion (say is (d/10,
1/10)-spread) for d n?/2, we can pick L
using n? random bits.
29
• GLW If H is an (n, d, vd)-expander,
• and L is (d/10, 1/10)-spread,
• then the distortion of X(H,L) is
exp(logdn)
• Picking d n? we get distortion exp(1/?) O(1)

Suffices to show For unit vector x ? X(H,L)
set W of lt n/20 vertices
30
Belief / Mass propagation
• Define Z z ? W z has gt d/10 neighbors in W
• By local (d/10, 1/10)-spread, mass in W \ Z
leaks out

It follows that
By expander mixing lemma, Z lt
W/d Iterating this logd n times
Completely analogous to iterative decoding of
binary codes, which extends to error-correction
over Reals. Alon This myopic analysis cannot
be improved! OPEN Fully explicit Euclidean
sections
31
Summary
• TCS goes hand in hand with Geometry

• Analysis

• Algebra

• Group Theory

• Number Theory
• Game
Theory

• Algebraic Geometry

• Topology
•
• Algorithmic/computational problems need math
tools, but also bring out new math problems and
techniques