Random Vectors, Random Matrices, and Diagrammatic Fun

Transcription

1 Random Vectors, Random Matrices, and Diagrammatic Fun Cristopher Moore University of New Mexico & Santa Fe Institute joint work with Alexander Russell University of Connecticut

2 A product of inner products Given G=(V,E) and a complex vector x v for each v, consider the product (u,v) E 1 x u,x v e.g. x 1,x 2 x 2,x 3 x 3,x 1 x 3,x 4 2

3 Now, with random vectors with x v uniform on the complex k-sphere, q(g; k) = Exp {x v } (u,v) E x u,x v what is q(g;k)? How hard is it to compute? Note: q(g;k)=0 unless G is Eulerian; otherwise rotate x v by a random phase x, y = k i=1 x i y i

4 Circuit partitions if there are rt partitions of edges into t cycles, the Circuit Partition Polynomial is j(g; z) = t=1 r t z t e.g. j(g; z) =z + z 2 #P-complete under Turing reductions

5 From integrals to combinatorics Theorem: q(g; k) = ( v V (k 1)! (k + d v 1)! ) j(g; k). Corollary: q(g;k) is #P-complete too Why this identity?

6 Outer products and tensors Inner product is a contraction of tensors: α 1 γ η β δ q(g; k) = x 1,x 2 x 2,x 3 x 3,x 1 x 3,x 4 2 = x 1 x 1 γ α x 2 x 2 α β x 3 x 3 x 3 x 3 βη γδ x 4 x 4 δ η. Einstein convention: tr M = Mα α, (UV) α γ = Uβ α Vγ β

7 A sum over permutations Exp v v v = 1 k 1 = 1 k δα α Exp v v v v v = 1 k(k + 1) ( ) δα α δ β β + δα β δα β + Exp x x d x d (k 1)! = (k + d 1)! π S d π Proof: commutes with any π and has trace 1

8 Wiring the vertices Averaging over xv turns each v into a sum over matchings of incoming and outgoing edges Sum of diagrams, each a cycle cover

9 Tracing the cycles Recall the vectors xv are k-dimensional For each cycle, k choices of basis vector v A diagram with t cycles contributes k t, so q(g; k) j(g; k) Real vectors undirected graphs #P-complete?

10 Determinant and Permanent Let be a matrix with entries in A n n {0, 1} det A = π S n ( 1) π i A i,πi perm A = π S n i A i,πi det A : geometric meaning, basis-independent, homomorphic, easy perm A : combinatorial, basis-dependent, hard

11 Complexity 0-1 PERMANENT is #P-complete [Valiant 79] However, it can be approximated using a rapidly-mixing Markov chain that samples random perfect matchings [Jerrum, Sinclair, Vigoda 04] Is there another approach, which is purely algebraic?

12 Godsil-Gutman estimator Let be a matrix with entries in A n n {0, 1} Let for uniformly random M ij = γ ij A ij γ ij {±1} Then E [ (det M) 2] = perm A

13 What?! It s true and, better yet, simple: E [ (det M) 2] = E π,σ ( 1) πσ i M i,πi M i,σi only contributing terms appear when π = σ and all M i,πi are nonzero: = π ( 1) ππ ( i M i,πi ) 2 = π A i,πi i

14 But what s the variance? For matrix A, define X = (det M) 2. Then E[X] = perm A How many samples do we need? Chebyshev: t E[X2 ] E[X] 2 Can we control this ratio?

15 Sadly... [KKLLL 93]: E[X 2 ] E[X] 2 =3n/2 poly(n) But, they show that if γ ij is uniform on the unit circle, or even just the cube roots of 1, E[X 2 ] E[X] 2 =2n/2 poly(n) Can we do better?

16 Higher-dimensional algebras? Barvinok: what if the γ ij are quaternions? Or higher-dimensional objects? [Chien, Rasmussen, Sinclair 03] In the Clifford algebra of dimension d, ( ( d) ) n/2 1 1+O In particular, for the quaternions, ( 3 2 ) n/2 Bad news: for d>2, we don t have an algorithm!

17 How to define the determinant? In the nonabelian case, order matters The conventional determinant takes each product from top to bottom: no efficient algorithm is known! [Barvinok 00]: symmetrize each term: sdet M = 1 n! π,α S n ( 1) π O(n d ) algorithm in a d-dimensional algebra i M αi,παi

18 Algebraic estimators Define M ij = ρ ij A ij, where ρ ij is drawn from some distribution on a nonabelian algebra A The Haar measure on unitary d d matrices The Gaussian measure on (independent entries) d d matrices det M takes values in A Define or X = det M 2 X = tr det M 2

19 Our results Two ways to get a scalar estimator: X = tr det M 2 X s = tr sdet M 2 We establish the following ratios: E [ X 2] E [X] 2 = ( ( d) ) n 1 1+O E[Xs 2 ( ] 2 n E[X s ] 2 = Ω n d ) Ratios differ by O(d 4 ) for X = det M 2

20 Permuted products E {ρij }[X s ] = κ A E {σi } E α,β ( tr i σ αi )( tr i σ βi ) = a d perm A where a d = E {σi } E α,β ( tr i σ αi )( tr i σ βi ) Covariance between a product of random matrices σ i taken in two different orders

21 The Cupcap Cometh What, for instance, is E σ1,σ 2,σ 3 (tr σ 1 σ 2 σ 3 )(tr σ 1 σ 3 σ 2 )? Both Haar and Gauss: E[σ i j (σk l ) ]= 1 d δik δ jl Diagrammatically: E[σ σ ]= 1 d

22 Diagrams and loops Form product by weaving matrices, and connect with cupcaps Tracing gives a factor of d for each loop σ 1 σ 2 σ 3 σ 1 σ 3 σ 2 E σ1,σ 2,σ 3 (tr σ 1 σ 2 σ 3 )(tr σ 1 σ 3 σ 2 ) = 1 d 2

23 A generating function Averaging over all permutations gives where and c(π) r = (1 2 n) a d = 1 d n E α d c([α,r]) is the number of cycles is a rotation Using Fourier analysis on S n, α α π a d = 1 d n (( ) n + d n +1 ( d n +1 ))

24 Fourier analysis First we write Then is an inner product a d = 1 d n E r,r d c(rr ) a d d c( ),P P where P is the uniform distribution on n-cycles is the trace of a combinatorial representation: action of on strings of length over d c( ) S n n {1,..., d} Fourier coefficients are Kostka numbers P is supported on hooks k The k-hook

25 The second moment The fourth moment is a sum of cupcaps: E σ [σ σ σ σ ]= 1 d 2 ( + ) Sum over double covers in which each appears 0, 2, or 4 times ρ ij More work, but similar ideas

26 The second moment Now the second moment can be bounded in terms of E π,σ S2n d c(π 1 (r,r)πσ 1 (r,r)σ) Complicated distribution of pairs of n-cycles in S 2n w α β γ α β γ w δ δ Bound in terms of uniform distribution Littlewood-Richardson rule: restrictions of irreps of S 2n to the Young subgroup S n S n

27 Oh, the irony We have a family of estimators with ratio (1+ε) n, but with no efficient algorithm We have a family of estimators that we can compute efficiently, but with ratio ~ 2 n Is there algebraic estimator which is both concentrated and efficiently computable?

28 Permuted products in finite groups If σ is an irreducible representation of G, E[σ σ ]= 1 d Let G be highly nonabelian or quasirandom : smallest irrep has large d Then for most permutations σ the pair (g 1 g 2 g t,g σ(1) g σ(2) g σ(t) ) is close to uniform in G G E.g. multiply by (g,g 1 ) σ reverses order

29 Conclusion Where random vectors random matrices random group elements are concerned, diagrammatic methods can turn averages/integrals into combinatorial problems... which can often be solved with Fourier analysis

30 An advertisement A representation of a group G is a homomorphism ψ from G to the unitary d d matrices... But what if there is no such thing? Theorem: dψ Pr [ψ(xy) =ψ(x) ψ(y)] d min No low-dimensional embeddings of G! Corollary: if f is a function from G to H and most of H s irreps have low dimension, Pr[f(xy) =f(x)f(y)] 1/ H No approximate homomorphisms from G to H

31 Another advertisement THE NATURE of COMPUTATION This book rocks! It's 900+ pages of awesome. It's a book I wish I had written, but now I'm saved the effort. You somehow manage to combine the fun of a popular book with the intellectual heft of a textbook. Scott Aaronson A treasure trove of information on algorithms and complexity, presented in the most delightful way. Vijay Vazirani Oxford University Press, 2011 Cristopher Moore Stephan Mertens