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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.
  • What about non-simple groups?
  • 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
Product Reingold-Vadhan-W
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
Start with a constant size H a d4, d, 1/4-graph.
  • 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
Start with G1 SYMd, T1 vd.
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
Start with G1 T1 Z2. Iterate Gn1
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
Spread subspaces
  • 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.
  • (t, ?)-spread ? (t, ?)-spread

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
Distortion/spread analysis
  • 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
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