Theoretical Computer Science methods in asymptotic geometry Avi Wigderson IAS, Princeton For Vitali Milman’s 70 th birthday
Feb 05, 2016
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