Advances in Random Matrix Theory: Let there be tools...3 Tools So many applications … Random matrix theory: catalyst for 21st century special functions, analytical techniques, statistical

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Advances in Random Matrix Theory: Let there be tools

Alan Edelman

MIT: Dept of Mathematics, Computer Science AI Laboratories

World Congress, Bernoulli Society Barcelona, Spain

Wednesday July 28, 20042/1/2005

Brian Sutton, Plamen Koev, Ioana Dumitriu,

Raj Rao and others

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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!

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Tools

So many applications … Random matrix theory: catalyst for 21st 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!

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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

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Precise statements require n→∞ etc.

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Let S = random symmetric Gaussian MATLAB: A=randn(n); S=(A+A’)/2;

S known as the Gaussian Orthogonal Ensemble n=20; s=30000; d=.05; e=[]; %gather up eigenvalues im=1; %imaginary(1) or real(0) for i=1:s,

v=eig(a)'; e=[e v]; end

axis('square'); axis([-1.5 1.5 -1 2]); hold on; t=-1:.01:1; plot(t,sqrt(1-t.^2),'r');

Wigner’s Semi-Circle The classical & most famous rand eig theorem

Normalized eigenvalue histogram is a semi-circle %matrix size, samples, sample dist

a=randn(n)+im*sqrt(-1)*randn(n);a=(a+a')/(2*sqrt(2*n*(im+1)));

hold off; [m x]=hist(e,-1.5:d:1.5); bar(x,m*pi/(2*d*n*s));

χ3χG

Gχ2

χ

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χ6

G

χ5

G χ6

χ4

G χ5

G χ4

2

χ3

χ1 G 1

Same eigenvalue distribution as GOE: O(n) storage !! O(n2) compute

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sym matrix to tridiagonal form

G4β

χ

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χ6β

G

χ5β

G χ6β

χ4β

G χ5β

χ3β

χ

χ2β

G 3β

χβ

G χ2β

G χβ

General beta beta:

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1: reals 2: complexes 4: quaternions

Bidiagonal Version corresponds To Wishart matrices of Statistics

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Tools Motivation: A condition number problem

The Polynomial Method 9 trick

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Jack & Hypergeometric of Matrix Argument MOPS: Ioana Dumitriu’s talk

The tridiagonal numerical 10

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Numerical Analysis: Condition Numbers

κ(A) = “condition number of A” If A=UΣ κ(A) = σmax/σmin . 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??? 2/1/2005

V’ is the svd, then

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Von Neumann & co. Solve Ax=b via x= (A’A) -1A’ b

M ≈A-1 Matrix Residual: ||AM-I||2 ||AM-I||2< 200κ2 n ε

How should we estimate κ?

Assume, as a model, that the elements of A areindependent standard normals!

≈

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Von Neumann & co. estimates (1947-1951)

“For a ‘random matrix’ of order n the expectation value has been shown to be about n”

Goldstine, von Neumann

√10n” Bargmann, Montgomery, vN

Goldstine, von Neumann

X ∞

P(κ

2 / 2 / 2 3

4 2 x x e x x y − − + =

Distribution of κ/n

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→∞

Experiment with n=2002/1/2005

Random cond numbers, n

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Finite n

n=10 n=25

n=50 n=100 2/1/2005

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Condition Number Distributions P(κ/n > x) ≈ 2/x P(κ/n2 > x) ≈ 4/x2

Real n x n, nÆ∞

Complex n x n, nÆ∞

Generalizations:

•β: 1=real, 2=complex

•finite matrices

•rectangular: m x n

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Real n x n, nÆ∞

Complex n x n, nÆ∞

Condition Number Distributions P(κ/n > x) ≈ 2/x P(κ/n2 > x) ≈ 4/x2

Square, nÆ∞: P(κ/nβ > x) ≈ (2ββ-1/Γ(β))/xβ (All Betas!!) General Formula: P(κ>x) ~Cµ/x β(n-m+1),

where µ= βWm-1,n+1 (β) and C is a known geometrical constant.

1F1((β/2)(n+1), ((β/2)(n+m-1); -(x/2)Im-1)

(n-m+1)/2th moment of the largest eigenvalue of

Density for the largest eig of W is known in terms of from which µ is available

Tracy-Widom law applies probably all beta for large m,n. Johnstone shows at least beta=1,2.

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Multivariate Orthogonal Polynomials &

The important special functions of the 21st century Begin with w(x) on I

∫ pκ(x)pλ(x) ∆(x)β ∏i w(xi)dxi = δκλ Jack Polynomials orthogonal for w=1

on the unit circle. Analogs of xm 2/1/2005

Hypergeometrics of Matrix Argument

Ioana Dumitriu’s talk

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Multivariate Hypergeometric Functions

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Multivariate Hypergeometric Functions

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Wishart (Laguerre) Matrices! 2/1/2005

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Plamen’s clever idea

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Mops (Dumitriu etc.) Symbolic

A=randn(n); S=(A+A’)/2; trace(S^4)

det(S^3)

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Symbolic MOPS applications

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β=3; hist(eig(S))

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Symbolic MOPS applications

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A=randn(m,n); hist(min(svd(A).^2))

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Smallest eigenvalue statistics

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9 trick

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Jack & Hypergeometric of Matrix Argument MOPS: Ioana Dumitriu’s talk The Polynomial Method -- Raj! The tridiagonal numerical 10

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Courtesy of the Polynomial Method 2/1/2005

RM Tool – Raj!

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∑ ∫ ∞

=

∞ −

= − Γ

= 1 0

1 1 1 ) (

1 ) ζ( k

s u

s

k du

e u

s s

6 π

4 1

3 1

2 1 1 ) ζ 2 (

2

2 2 2 = + + + + = …

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The Riemann Zeta Function

On the real line with x>1, for example

May be analytically extended to the complex plane, with singularity only at x=1.

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-3 -2 -1 0 ½ 1 2 3

∑ ∫ ∞

=

∞ −

= − Γ

= 1 0

1 1 1 ) (

1 ) ζ( k

x u

x

k du

e u

x x

All nontrivial roots of ζ(x) satisfy Re(x)=1/2. (Trivial roots at negative even integers.)

The Riemann Hypothesis

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-3 -2 -1 0 ½ 1 2 3

The Riemann Hypothesis

∑∫∞

=

∞ −

=−Γ

=10

1 11)(

1)ζ(k

xu

x

kdu

eu

xx

All nontrivial roots of ζ(x) satisfy Re(x)=1/2. (Trivial roots at negative even integers.)

ζ(x)| along Re(x)=1/2 Zeros =.5+i* 14.134725142 21.022039639 25.010857580 30.424876126 32.935061588 37.586178159 40.918719012 43.327073281 48.005150881 49.773832478 52.970321478 56.446247697 59.347044003

|

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Computation of Zeros 10^k+1

through 10^k+10,000 for k=12,21,22.

See http://www.research.att.com/~amo/zeta_tables/

A=randn(n)+i*randn(n); S=(A+A’)/2;

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Odlyzko’s fantastic computation of

Spacings behave like the eigenvalues of

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Nearest Neighbor Spacings & Pairwise Correlation Functions

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Painlevé Equations

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Spacings

Normalize so that average spacing = 1.

jth) Conjecture: These functions are the same for random

matrices and Riemann zeta

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Take a large collection of consecutive zeros/eigenvalues.

Spacing Function = Histogram of consecutive differences (the (k+1)st – the kth) Pairwise Correlation Function = Histogram of all possible differences (the kth – the

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Everyone’s Favorite Tridiagonal

-21 1

1-21 1-2

… ……

… …

1 n2

d2 dx2

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Everyone’s Favorite Tridiagonal

-21 1

1-21 1-2

… ……

… …

1 n2

d2 dx2

1 (βn)1/2+

G

G G

dW β1/2+

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Stochastic Operator Limit

,

N(0,2)χ χN(0,2)χ

χN(0,2)χ χN(0,2)

nβ2 1 ~H

β

β2β

2)β(n1)β(n

1)β(n

β n

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

−−

−

β 2 x

dx d

2

2

+−

,G β

2HH nn β n +≈

∞

… …

…

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, dW

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Largest Eigenvalue Plots

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MATLAB

beta=1; n=1e9; opts.disp=0;opts.issym=1;alpha=10; k=round(alpha*n^(1/3)); % cutoff parametersd=sqrt(chi2rnd( beta*(n:-1:(n-k-1))))';H=spdiags( d,1,k,k)+spdiags(randn(k,1),0,k,k);H=(H+H')/sqrt(4*n*beta);eigs(H,1,1,opts)

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Tricks to get O(n9) speedup • •

•

general purpose eigensolvers.) •

convergence of the largest eigenvalue) •

structures and numerical choices.) • 1/3

determined numerically by the top k × k segment of n. (This is aninteresting mathematical statement related to the decay of the Airyfunction.)

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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

The shift-and-invert accelerator to Lanczos and Arnoldi (Since we know the eigenvalues are near 1, we can accelerate the

The ARPACK software package as made available seamlessly in MATLAB (The Arnoldi package contains state of the art data

The observation that if k = 10n , then the largest eigenvalue is

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Level Densities

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Open Problems

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

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Random matrix tools!

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Advances in Random Matrix Theory: Let there be tools...3 Tools So many applications … Random matrix theory: catalyst for 21st century special functions, analytical techniques, statistical