Theoretical Computer Science methods in asymptotic geometry

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TheoreticalComputerScience methods in
asymptotic geometry

• Avi Wigderson
• IAS, Princeton
• For Vitali Milmans 70th birthday

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Three topicsMethods 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

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Parallel Repetition of Games and Periodic
Foams
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Isoperimetric problem Minimize surface
area given volume.
One bubble. Best solution Sphere
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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
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Our Problem
Minimum surface area of body tiling Rd with
period Zd ? d2 area
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gt4
Choe89 Optimal!
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Bounds in d dimensions
OPT
OPT
Spherical Cubes exist! Probabilistic
construction! (simpler analysis
Alon-Klartag) OPEN Explicit?
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Randomized Rounding
Round points in Rd to points in Zd such that for
every x,y 1. 2.
x
y
1
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Spine
Surface blocking all cycles that wrap around
Torus
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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.
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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!

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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
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Hardness amplification byparallel 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 ?
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Zig-zag Graph Product and Cayley expanders in
non-simple groups
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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
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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.
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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).

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Explicit Constructions (Combinatorial)-Zigzag
Product Reingold-Vadhan-W
K an n, m, ?-graph. H an m, d, ?-graph.
Edges
Combinatorial construction of expanders.
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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.
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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
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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.
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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 )
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Belief Propagation in Codes and L2 sections of L1
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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.

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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)

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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

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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

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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.
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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
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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
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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