Transcript The zigzag product, Expander graphs & Combinatorics vs

Theoretical
Computer
Science
methods in asymptotic geometry
Avi Wigderson
IAS, 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:
Choe’89:
>4
4
Optimal!
Bounds in d dimensions
≤OPT≤
[Kindler,O’Donnell,
≤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.
1
x y
Spine
Torus
Surface blocking all
cycles that wrap around
Probabilistic construction of spine
Step 1
Probabilistically
construct B, which in
expectation satisfies
Step 2
Sample independent
translations of B until
[0,1)d is covered, adding
new boundaries to spine.
B
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] >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
αj
Bob
βj
Check if αj + βj = bj
Pr [YES] ≤ 1-
Hardness amplification by
parallel repetition
2n variables:
X1,X2,…,Xn, Y1,Y2,…,Yn
m linear equations:
Xi1 + Yi1 = b1
Xi2 + Yi2 = b2
Game Gk
Draw j1,j2,…jk  [m] at random
Xij1Xij2 Xijk
Alice
Yij1Yij2 Yijk
Bob
…..
Xim + Yim = bm
αj1αj2 αjk
βj1βj2 βjk
Promise: no setting of the
Check if αjt + βjt = bjt t [k]
Xi,Yi satisfy more than
(1-)m of all equations
Pr[YES] ≤ (1-2)k [Raz,Holenstein,Rao]
[Feige-Kindler-O’Donnell]
Spherical Cubes 
X Pr[YES] ≥ (1-2)k
[KORW]Spherical Cubes 
[Raz]
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-PhilipsSarnak,…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.
G = SL2(p) : group 2 x 2 matrices of det 1 over Zp.
11
01
S = { M1 , M2 } : M 1 = ( ) , M2 = (
10
11
)
Theorem. [LPS] Cay(G,S) is an expander family.
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.
Definition. K z H has vertices {(v,h) : vK, hH}.
v
(v,h)
u
Edges
Thm. [RVW]
K z H is an [nm, d2, +]-graph,
K z H is an expander iff K and H are.
Combinatorial construction of expanders.
H
Iterative Construction of Expanders
K an [n,m,]-graph. H an [m,d,] -graph.
[RVW] K z H is an [nm,d2,+]-graph.
The construction: A sequence K1,K2,… of expanders
Start with a constant size H a [d4, d, 1/4]-graph.
• K1 = H2
• Ki+1 = Ki2 z H
[RVW] Ki is a [d4i, d2, ½]-graph.
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
d
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.
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
77]:
[Kashin 77, Figiel-Lindenstrauss-Milman
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 xX
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
For
dim(X)=N/2
X= ker(A)
[Arstein-Milman]
(X) = (√N||x||2)/||x||1 = O(1)
• [Kashin ’77, Garnaev-Gluskin ’84]
A a random sign matrix.
• [Arstein-Milman ’06]
Expander walk on A’s columns
# random bits
O(N2 )
• [Lovett-Sodin ‘07]
O(N log N)
O(N)
Expander walk + k-wise independence
• [Guruswami-Lee-W ’08] (X) = exp(1/)
Expander codes & “belief propagation”
N
>0
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)
Ingredients for X=X(H,L):
- H(V,E): a d-regular expander
- L  Rd : a random subspace
X(H,L) = {
xRE
: xE(v)  L v V }
Belongs to L
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
V
W
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”
It follows that
By expander mixing lemma,
Z
W
|Z| < |W|/d
Iterating this logd n times…
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