TheoreticalComputerScience methods in asymptotic geometry

Avi WigdersonIAS, Princeton

For Vitali Milman’s 70th birthday

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

Parallel Repetition of Games and Periodic Foams

Isoperimetric problem: Minimize surface area given volume.

One bubble. Best solution: Sphere

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

Our Problem

Minimum surface area of body tiling Rd with period Zd ?

d=2 area:

4>4Choe’89:Optimal!

Bounds in d dimensions

≤ OPT ≤

[Kindler,O’Donn[Kindler,O’Donnell,ell, Rao,Wigderson]Rao,Wigderson] ≤OPT≤

“Spherical Cubes” exist!Probabilistic construction!(simpler analysis [Alon-Klartag])

OPEN: Explicit?

Randomized Rounding

Round points in Rd to points in Zd

such that for every x,y

1.

2.

x y1

Spine

TorusSurface blocking allcycles that wrap around

Probabilistic construction of spine

Step 1

Probabilisticallyconstruct B, which in expectation satisfies

BB

Step 2

Sample independent translations of B until [0,1)d is covered, adding new boundaries to spine.

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 elimination2) Does there exist z satisfying ≥ .9m equations? NP-hard – PCP Theorem [AS,ALMSS]3) Does there exist z satisfying ≥ .5m equations? Easy – YES!

[Hastad] >0, it is NP-hard to distinguish (A,b) which are not (½+)-satisfiable, from those (1-)-satisfiable!

Linear equations as Games

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

Game G

Draw j [m] at random

Xij Yij Alice Bob

αj βj

Check if αj + βj = bj

Pr [YES] ≤ 1-

Hardness amplification byparallel repetition

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

Game Gk

Draw j1,j2,…jk [m] at random

Xij1Xij2 Xijk Yij1Yij2 Yijk Alice Bob

αj1αj2 αjk βj1βj2 βjk

Check if αjt + βjt = bjt t [k]

Pr[YES] ≤ (1-2)k

[Raz,Holenstein,Rao] Pr[YES] ≥ (1-2)k

[Feige-Kindler-O’Donnell] Spherical Cubes

[Raz]X[KORW]Spherical Cubes

Zig-zag Graph Product and Cayley expanders in

non-simple groups

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) <1

Algebraic explicit constructions [Margulis ‘73,Gaber-Galil,Alon-Milman,Lubotzky-Philips-Sarnak,…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 SS-1.

Theorem. [LPS] Cay(G,S) is an expander family.

G = SL2(p) : group 2 x 2 matrices of det 1 over Zp.

S = { M1 , M2 } : M1 = ( ) , M2 = ( ) 1 10 1

1 01 1

Algebraic Constructions (cont.)

[Margulis] SLn(p) is expanding (n≥3 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 >log n generators - k-solvable gps of size n require >log(k)n gens [LW] -Some p-groups (eg SL3(pZ)/SL3(pnZ) ) expand with O(1) generating sets (again relies on property T).

Explicit Constructions (Combinatorial)-Zigzag Product [Reingold-Vadhan-W]

K an [n, m, ]-graph. H an [m, d, ]-graph.

Combinatorial construction of expanders.

H

v u(v,h)

Thm. [RVW] K z H is an [nm, d2, +]-graph,

Definition. K z H has vertices {(v,h) : vK, hH}.

K z H is an expander iff K and H are.

Edges

Iterative Construction of Expanders

K an [n,m,]-graph. H an [m,d,] -graph.

The construction: A sequence K1,K2,… of expandersStart with a constant size H a [d4, d, 1/4]-graph.

• K1 = H2

[RVW] Ki is a [d4i, d2, ½]-graph.

[RVW] K z H is an [nm,d2,+]-graph.

• Ki+1 = Ki2 z H

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 <T> = B, <S> = A , S = sB (S is a single B-orbit)[Alon-Lubotzky-W] Cay(A x B, TsT ) = Cay (A,S) z Cay(B,T)

[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 Gn+1,Tn+1 from Gn,Tn

Constant degree expansion in iterated wreath-products [Rosenman-Shalev-W]

Start with G1 = SYMd, |T1| ≤ √d. [Kassabov]

Iterate: Gn+1 = 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 Gn

d

Theorem [RSW] Cay(Gn, Tn) constant degree expanders.

d

n

Near-constant degree expansion in solvable groups [Meshulam-W]

Start with G1 = T1 = Z2. Iterate: Gn+1 = Gn x Fp[Gn]

Get (G1 ,T1 ), (G2 ,T2),…, (Gn ,Tn ),...

Cay(Gn,Tn ) expands few expanding orbits for Fp[Gn]

Conjecture (true for Gn’s): 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] )

Belief Propagation in Codes and L2 sections of L1

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 |x|1 = Θ(√N)|x|2 xX

L2 mass of x is spread across many coordinates #{ i : |xi| ~ √N||x||2 } = Ω(N)

• Analogy: error-correcting codes: Subspace C of F2N

with every nonzero c C has (N) Hamming weight.

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 L2~L1

X RN dim(X) vs. istortion(X) (X) = Maxx X(√N||x||2)/||x||1

We focus on: dim(X)=(N) & (X) =O(1)

Derandomization results [Arstein-Milman]For

dim(X)=N/2 (X) = (√N||x||2)/||x||1 = 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 A’s columns

• [Lovett-Sodin ‘07] O(N)

Expander walk + k-wise independence

• [Guruswami-Lee-W ’08] (X) = exp(1/) N >0

Expander codes & “belief propagation”

Spread subspaces

Key ideas [Guruswami-Lee-Razborov]: L Rd is (t,)-spread if every x L, S [d], |S|≤t ||xS||2 ≤ (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

Constant distortion construction [GLW](like Tanner codes)

Belongs to L

Ingredients for X=X(H,L):

- H(V,E): a d-regular expander- L Rd : a random subspace

X(H,L) = { xRE : 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.

Distortion/spread analysis [GLW]: If H is an (n, d, √d)-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 < n/20 vertices

WV

Belief / Mass propagation• Define Z = { z W : z has > d/10 neighbors in W }• By local (d/10, 1/10)-spread, mass in W \ Z “leaks

out”

By expander mixing lemma,

|Z| < |W|/d

Iterating this logd n times…

It follows that

WZ

V

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

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