Advances in Random Matrix Theory: Let there be tools

Transcription

1 Advances in Random Matrix Theory: Let there be tools Alan Edelman Brian Sutton, Plamen Koev, Ioana Dumitriu, Raj Rao and others MIT: Dept of Mathematics, Computer Science AI Laboratories World Congress, Bernoulli Society Barcelona, Spain Wednesday July 28, 2004

2 Message Ingredient: Take Any important mathematics Then Randomize! This will have many applications! We can t keep this in the hands of specialists anymore: Need tools! 2

3 Tools So many applications Random matrix theory: catalyst for 2 st century special functions, analytical techniques, statistical techniques In addition to mathematics and papers Need tools for the novice! Need tools for the engineers! Need tools for the specialists! 3

4 Themes of this talk Tools for general beta What is beta? Think of it as a measure of (inverse) volatility in classical random matrices. Tools for complicated derived random matrices Tools for numerical computation and simulation 4

5 Wigner s Semi-Circle The classical & most famous rand eig theorem Let S = random symmetric Gaussian MATLAB: A=randn(n); S=(A+A )/2; S known as the Gaussian Orthogonal Ensemble Normalized eigenvalue histogram is a semi-circle n=20; s=30000; d=.05; %matrix size, samples, sample dist e=[]; %gather up eigenvalues im=; %imaginary() or real(0) for i=:s, a=randn(n)+im*sqrt(-)*randn(n);a=(a+a')/(2*sqrt(2*n*(im+))); v=eig(a)'; e=[e v]; end hold off; [m x]=hist(e,-.5:d:.5); bar(x,m*pi/(2*d*n*s)); axis('square'); axis([ ]); hold on; t=-:.0:; plot(t,sqrt(-t.^2),'r'); Precise statements require n etc. 5

6 sym matrix to tridiagonal form χ 4 χ 3 Same eigenvalue distribution χ G as 3 χgoe: O(n) storage!! O(n 2 ) compute 2 G χ 6 χ 6 G χ 5 χ 5 G χ 4 G χ 2 G χ χ G 6

7 General beta G χ 6β χ 6β G beta: : reals 2: complexes 4: quaternions χ 5β χ 5β G χ Bidiagonal 4β Version corresponds χ To Wishart G 4β χmatrices of Statistics 3β χ G 3β χ 2β χ 2β G χ β χ β G 7

8 Tools Motivation: A condition number problem Jack & Hypergeometric of Matrix Argument MOPS: Ioana Dumitriu s talk The Polynomial Method The tridiagonal numerical 09 trick 8

9 Tools Motivation: A condition number problem Jack & Hypergeometric of Matrix Argument MOPS: Ioana Dumitriu s talk The Polynomial Method The tridiagonal numerical 09 trick 9

10 Numerical Analysis: Condition Numbers κ(a) = condition number of A If A=UΣV is the svd, then κ(a) = σ max /σ min. Alternatively, κ(a) = λ max (A A)/ λ min (A A) One number that measures digits lost in finite precision and general matrix badness Small=good Large=bad The condition of a random matrix??? 0

11 Von Neumann & co. Solve Ax=b via x= (A A) - A b M A - Matrix Residual: AM-I 2 AM-I 2 < 200κ 2 n ε How should we estimate κ? Assume, as a model, that the elements of A are independent standard normals!

12 Von Neumann & co. estimates (947-95) For a random matrix of order n the expectation value has been shown to be about n X Goldstine, von Neumann we choose two different values of κ, namely n and 0n P(κ<n) 0.02 P(κ< 0n) 0.44 Bargmann, Montgomery, vn With a probability ~ κ < 0n P(κ<0n) 0.80 Goldstine, von Neumann 2

13 Random cond numbers, n y Distribution of κ/n = x e 2 / x 2 / x 3 x 2 Experiment with n=200 3

14 Finite n n=0 n=25 n=50 n=00 4

15 Condition Number Distributions P(κ/n > x) 2/x P(κ/n 2 > x) 4/x 2 Real n x n, næ Generalizations: β: =real, 2=complex finite matrices rectangular: m x n Complex n x n, næ 5

16 Condition Number Distributions P(κ/n > x) 2/x P(κ/n 2 > x) 4/x 2 Square, næ : P(κ/n β > x) (2β β- /Γ(β))/x β (All Betas!!) General Real Formula: n x n, næ P(κ>x) ~Cµ/x β(n-m+), where µ= β(n-m+)/2th moment of the largest eigenvalue of W m-,n+ (β) and C is a known geometrical constant. Complex n x n, næ Density for the largest eig of W is known in terms of F ((β/2)(n+), ((β/2)(n+m-); -(x/2)i m- ) from which µ is available Tracy-Widom law applies probably all beta for large m,n. Johnstone shows at least beta=,2. 6

17 Tools Motivation: A condition number problem Jack & Hypergeometric of Matrix Argument MOPS: Ioana Dumitriu s talk The Polynomial Method The tridiagonal numerical 09 trick 7

18 Multivariate Orthogonal Polynomials & Hypergeometrics of Matrix Argument Ioana Dumitriu s talk The important special functions of the 2 st century Begin with w(x) on I p κ (x)p λ (x) (x) β i w(x i )dx i = δ κλ Jack Polynomials orthogonal for w= on the unit circle. Analogs of x m 8

19 Multivariate Hypergeometric Functions 9

20 Multivariate Hypergeometric Functions 20

21 Wishart (Laguerre) Matrices! 2

22 Plamen s clever idea 22

23 Tools Motivation: A condition number problem Jack & Hypergeometric of Matrix Argument MOPS: Ioana Dumitriu s talk The Polynomial Method The tridiagonal numerical 09 trick 23

24 Mops (Dumitriu etc.) Symbolic 24

25 Symbolic MOPS applications A=randn(n); S=(A+A )/2; trace(s^4) det(s^3) 25

26 Symbolic MOPS applications β=3; hist(eig(s)) 26

27 Smallest eigenvalue statistics A=randn(m,n); hist(min(svd(a).^2)) 27

28 Tools Motivation: A condition number problem Jack & Hypergeometric of Matrix Argument MOPS: Ioana Dumitriu s talk The Polynomial Method -- Raj! The tridiagonal numerical 09 trick 28

29 RM Tool Raj! Courtesy of the Polynomial Method 29

30 30

31 = = Γ = 0 ) ( ) ζ( k s u s k du e u s s 6 π ) ζ 2 ( = = 3 The Riemann Zeta Function On the real line with x>, for example May be analytically extended to the complex plane, with singularity only at x=.

32 ½ 2 3 = = Γ = 0 ) ( ) ζ( k x u x k du e u x x All nontrivial roots of ζ(x) satisfy Re(x)=/2. (Trivial roots at negative even integers.) The Riemann Hypothesis

33 ζ( x) ζ(x) along Re(x)=/2 Zeros =.5+i* The Riemann Hypothesis u x = du = Γ( x) e u k x k= ½ All nontrivial roots of ζ(x) satisfy Re(x)=/2. (Trivial roots at negative even integers.) 33

34 Computation of Zeros Odlyzko s fantastic computation of 0^k+ through 0^k+0,000 for k=2,2,22. See Spacings behave like the eigenvalues of A=randn(n)+i*randn(n); S=(A+A )/2; 34

35 Nearest Neighbor Spacings & Pairwise Correlation Functions 35

36 Painlevé Equations 36

37 Spacings Take a large collection of consecutive zeros/eigenvalues. Normalize so that average spacing =. Spacing Function = Histogram of consecutive differences (the (k+)st the kth) Pairwise Correlation Function = Histogram of all possible differences (the kth the jth) Conjecture: These functions are the same for random matrices and Riemann zeta 37

38 Tools Motivation: A condition number problem Jack & Hypergeometric of Matrix Argument MOPS: Ioana Dumitriu s talk The Polynomial Method The tridiagonal numerical 09 trick 38

39 Everyone s Favorite Tridiagonal n d 2 dx 2 39

40 Everyone s Favorite Tridiagonal n (βn) /2 G G G d 2 dx 2 + dw β /2 40

41 4 Stochastic Operator Limit, N(0,2) χ χ N(0,2) χ χ N(0,2) χ χ N(0,2) nβ 2 ~ H β β 2β 2)β (n )β (n )β (n β n β 2 x dx d 2 2 +, G β 2 H H n n β n + dw,

42 Largest Eigenvalue Plots 42

43 MATLAB beta=; n=e9; opts.disp=0;opts.issym=; alpha=0; k=round(alpha*n^(/3)); % cutoff parameters d=sqrt(chi2rnd( beta*(n:-:(n-k-))))'; H=spdiags( d,,k,k)+spdiags(randn(k,),0,k,k); H=(H+H')/sqrt(4*n*beta); eigs(h,,,opts) 43

44 Tricks to get O(n 9 ) speedup Sparse matrix storage (Only O(n) storage is used) Tridiagonal Ensemble Formulas (Any beta is available due to the tridiagonal ensemble) The Lanczos Algorithm for Eigenvalue Computation ( This allows the computation of the extreme eigenvalue faster than typical general purpose eigensolvers.) The shift-and-invert accelerator to Lanczos and Arnoldi (Since we know the eigenvalues are near, we can accelerate the convergence of the largest eigenvalue) The ARPACK software package as made available seamlessly in MATLAB (The Arnoldi package contains state of the art data structures and numerical choices.) The observation that if k = 0n /3, then the largest eigenvalue is determined numerically by the top k k segment of n. (This is an interesting mathematical statement related to the decay of the Airy function.) 44

45 Level Densities 45

46 Open Problems The distribution for general beta Seems to be governed by a convection-diffusion equation 46

47 Random matrix tools! 47