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T heoretical C omputer S cience methods in asymptotic geometry

Avi Wigderson IAS, Princeton For Vitali Milman’s 70 th birthday. T heoretical C omputer S cience methods in asymptotic geometry. Three topics: Methods and Applications. Parallel Repetition of games and Periodic foams Zig-zag Graph Product and

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T heoretical C omputer S cience methods in asymptotic geometry

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  1. Avi Wigderson IAS, Princeton For Vitali Milman’s 70th birthday TheoreticalComputerScience methods in asymptotic geometry

  2. Three topics:MethodsandApplications • 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-Phelan1994 Even better

  6. Our Problem Minimum surface area of body tiling Rd with period Zd ? d=2 area: Choe’89: Optimal! >4 4

  7. [Kindler,O’Donnell, Rao,Wigderson] 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] >0, 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 αj βj Check if αj +βj = 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 byparallel repetition Game Gk Draw j1,j2,…jk  [m] at random Xij1Xij2 XijkYij1Yij2 Yijk Alice Bob αj1αj2 αjkβj1βj2 βjk Check if αjt +βjt = bjt t [k] Pr[YES] ≤ (1-2)k [Raz,Holenstein,Rao] Pr[YES] ≥ (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-O’Donnell] Spherical Cubes  X [Raz] [KORW]Spherical 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) <1

  16. G= SL2(p) : group 2 x 2 matrices of det 1 over Zp. S= { M1 , M2 } : M1 = ( ) , M2 = ( ) 1 1 0 1 1 0 1 1 Algebraic explicit constructions [Margulis ‘73,Gaber-Galil,Alon-Milman,Lubotzky-Philips-Sarnak,…Nikolov,Kassabov,…,Bourgain-Gamburd ‘09,…] Many such constructions are Cayleygraphs. Ga finite group, Sa set of generators. Def. Cay(G,S) has vertices Gand 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 (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 Snis 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).

  18. H Definition. K zHhas vertices {(v,h) : vK, hH}. (v,h) v u Thm. [RVW] K zHis an [nm, d2, +]-graph, K zH is an expander iff Kand Hare. Explicit Constructions (Combinatorial)-Zigzag Product [Reingold-Vadhan-W] Kan [n, m, ]-graph. H an [m, d, ]-graph. Edges Combinatorial construction of expanders.

  19. [RVW]Kz His an [nm,d2,+]-graph. • Ki+1 = Ki2z H Iterative Construction of Expanders Kan [n,m,]-graph. Han [m,d,] -graph. The construction: A sequence K1,K2,… of expanders Start with a constant size Ha [d4, d, 1/4]-graph. • K1 = H2 [RVW]Ki is a [d4i, d2, ½]-graph.

  20. [Alon-Lubotzky-W] Cay(Ax B, TsT) = Cay (A,S) z Cay(B,T) Semi-direct Product of groups A,Bgroups. Bacts on A. Semi-direct product: Ax B Connection: semi-direct product is a special case of zigzag Assume <T> = B, <S> = A, S= sB(Sis 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 Gn+1,Tn+1 fromGn,Tn

  21. 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 Gnd d n Theorem[RSW]Cay(Gn, Tn) constant degree expanders.

  22. 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  Ghas ≤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 randomsubspaceX RNwith 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 Cof F2Nwith 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 L2~L1 X RNdim(X)vs.istortion(X) (X) =Maxx X(√N||x||2)/||x||1 We focus on: dim(X)=(N) & (X) =O(1)

  26. 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”

  27. Spread subspaces Key ideas [Guruswami-Lee-Razborov]: L  Rdis (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

  28. 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 } 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. Belongs to L

  29. Distortion/spread analysis [GLW]: If H is an (n, d,√d)-expander, and Lis (d/10, 1/10)-spread, then the distortion of X(H,L) is exp(logdn) Pickingd = n we get distortion exp(1/) =O(1) Suffices to show: For unit vector x  X(H,L) & setWof<n/20vertices W V

  30. Belief / Mass propagation • Define Z= { zW:zhas>d/10neighbors inW} • By local(d/10, 1/10)-spread, mass in W \ Z “leaks out” It follows that By expander mixing lemma, |Z| < |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 Z W V

  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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