Progress in the method of Ghosts and Shadows for Beta Ensembles

Progress in the method of Ghosts and Shadows for Beta Ensembles Alan Edelman (MIT) Alex Dubbs (MIT) and Plamen Koev (SJS) Oct 8, 2012 1/47

Wishart Matrices (arbitrary covariance) G=mxn matrix of Gaussians Σ=mxn semidefinite matrix G G Σ is similar to A=Σ ½ G GΣ ½ For β=1,2,4, the joint eigenvalue density of A has a formula: Known for β=2 in some circles as Harish Chandra Itzykson Zuber 2/47

Main Purpose of this talk Eigenvalue density of G G Σ ( similar to A=Σ ½ G GΣ ½ ) Present an algorithm for sampling from this density Show how the method of Ghosts and Shadows can be used to derive this algorithm Further evidence that β=1,2,4 need not be special 3/47

Eigenvalues of GOE (β=1) Naïve Way: MATLAB : A=randn(n); S=(A+A )/sqrt(2*n);eig(s) R: A=matrix(rnorm(n*n),ncol=n);S=(a+t(a))/sqrt(2*n);eigen(S,symmetric=T,only.values=T)$values; Mathematica: A=RandomArray[NormalDistribution[],{n,n}];S=(A+Transpose[A])/Sqrt[n];Eigenvalues[s] 4/47

Eigenvalues of GOE/GUE/GSE and beyond Real Sym. Tridiagonal: β=1:real, β=2:complex, β=4:quaternion Diagonals N(0,2), Off diagonals chi random variables General β>0 Computations significantly faster 5/47

Introduction to Ghosts G 1 is a standard normal N(0,1) G 2 is a complex normal (G 1 +ig 1 ) G 4 is a quaternion normal (G 1 +ig 1 +jg 1 +kg 1 ) G β (β>0) seems to often work just fine Ghost Gaussian 6/47

Chi squared 2 Defn: χ β is the sum of β iid squares of standard normals if β=1,2, Generalizes for non integer β as the gamma function interpolates factorial χ β is the sqrt of the sum of squares (which generalizes) (wikipedia chi distriubtion) G 1 is χ 1, G 2 is χ 2, G 4 is χ 4 So why not G β is χ β? I call χ β the shadow of G β 7/47

Linear Algebra Example Given a ghost Gaussian, G β, G β is a real chibeta variable χ β variable. Gram Schmidt: G β G β G β χ 3β G β G β χ 3β G β G β χ 3β G β G β G β G β G β G β G β χ 2β G β H 3 H 2 H 1 χ 2β G β G β G β G β G β G β G β Χ β or G β G β G β G β G β G β G β G β G β = [Q] * χ 3β G β G β χ 2β G β Χ β 8/47

Linear Algebra Example Given a ghost Gaussian, G β, G β is a real chibeta variable χ β variable. Gram Schmidt: G β G β G β χ 3β G β G β χ 3β G β G β χ 3β G β G β G β G β G β G β G β χ 2β G β H 3 H 2 H 1 χ 2β G β G β G β G β G β G β G β Χ β or G β G β G β G β G β G β G β G β G β = [Q] * χ 3β G β G β χ 2β G β Χ β Q has β Haar Measure! We have computed moments! 9/47

Tridiagonalizing Example G β G β G β G β Symmetric Part of G β G β G β G β G β G β G β G β G β G β G β G β G β (Silverstein, Trotter, etc) 10/47

Histogram without Histogramming: Sturm Sequences Count #eigs < 0.5: Count sign changes in Det( (A 0.5*I)[1:k,1:k] ) Count #eigs in [x,x+h] Take difference in number of sign changes at x+h and x Mentioned in Dumitriu and E 2006, Used theoretically in Albrecht, Chan, and E 2008 11/47

A good computational trick is a good theoretical trick! 12/47

Stochastic Differential Eigen Equations Tridiagonal Models Suggest SDE s with Brownian motion as infinite limit: E: 2002 E and Sutton 2005, 2007 Brian Rider and (Ramirez, Cambronero, Virag, etc.) (Lots of beautiful results!) (Not today s talk) 13/47

Tracy Widom Computations Eigenvalues without the whole matrix! Never construct the entire tridiagonal matrix! Just say the upper 10n 1/3 by 10n 1/3 Compute largest eigenvalue of that, perhaps using Lanczos with shift and invert strategy! Can compute for n amazingly large!!!!! And any beta. E: 2003 Persson and E: 2005 14/47

Other Computational Results Infinite Random Matrix Theory The Free Probability Calculator (Raj) Finite Random Matrix Theory MOPS (Ioana Dumitriu) (β: orthogonal polynomials) Hypergeometrics of Matrix Argument (Plamen Koev) (β: distribution functions for finite stats such as the finite Tracy Widom laws for Laguerre) Good stuff, but not today. 15/47

Scary Ideas in Mathematics Zero Negative Radical Irrational Imaginary Ghosts: Something like a commutative algebra of random variables that generalizes random Reals, Complexes, and Quaternions and inspires theoretical results and numerical computation 16/47

Did you say commutative?? Quaternions don t commute. Yes but random quaternions do! If x and y are G 4 then x*y and y*x are identically distributed! 17/47

Hermite: RMT Densities c λ i λ j β e λ i 2 /2 (Gaussian Ensemble) Laguerre: Jacobi: c λ i λ j β λ m i e λ i (Wishart Matrices) c λ i λ j β λ m 1 i (1 λi ) m 2 (Manova Matrices) Fourier: c λ i λ j β (on the complex unit circle) Jack Polynomials Traditional Story: Count the real parameters β=1,2,4 for real, complex, quaternion Applications: β=1: All of Multivariate Statistics β=2:tons, β=4: Almost nobody cares Dyson 1962 Threefold Way Three Associative Division Rings 18/47

β Ghosts β=1 has one real Gaussian (G) β=2 has two real Gaussians (G+iG) β=4 has four real Gaussians (G+iG+jG+kG) β 1 has one real part and (β 1) Ghost parts 19/47

Introductory Theory There is an advanced theory emerging A β ghost is a spherically symmetric random variable defined on R β A shadow is a derived real or complex quantity 20/47

Wishart Matrices (arbitrary covariance) G=mxn matrix of Gaussians Σ=mxn semidefinite matrix G G Σ is similar to A=Σ ½ G GΣ ½ For β=1,2,4, the joint eigenvalue density of A has a formula: 21/47

Joint Eigenvalue density of G G Σ The 0 F 0 function is a hypergeometric function of two matrix arguments that depends only on the eigenvalues of the matrices. Formulas and software exist. 22/47

Laguerre: Generalization of Laguerre Versus Wishart: 23/47

General β? The joint density: is a probability density for all β>0. Goals: Algorithm for sampling from this density Get a feel for the density s ghost meaning 24/47

Main Result An algorithm derived from ghosts that samples eigenvalues A MATLAB implementation that is consistent with other beta ized formulas Largest Eigenvalue Smallest Eigenvalue 25/47

Real quantity Working with Ghosts 26/47

More practice with Ghosts 27/47

Bidiagonalizing Σ=I Z Z has the Σ=I density giving a special case of 28/47

The Algorithm for Z=GΣ ½ 29/47

The Algorithm for Z=GΣ ½ 30/47

Removing U and V 31/47

Algorithm cont. 32/47

Completion of Recursion 33/47

Numerical Experiments Largest Eigenvalue Analytic Formula for largest eigenvalue dist E and Koev: software to compute 34/47

1 m3n3beta5.000m150.stag.a.fig 0.9 0.8 0.7 0.6 F(x) 0.5 0.4 0.3 0.2 0.1 0 0 20 40 60 80 100 120 x 35 35/47

1 m4n4beta2.500m130.stag.a.fig 0.9 0.8 0.7 0.6 F(x) 0.5 0.4 0.3 0.2 0.1 0 0 10 20 30 40 50 60 70 80 90 100 x 36/47

1 m5n4beta0.750m120.1234.a.fig 0.9 0.8 0.7 0.6 F(x) 0.5 0.4 0.3 0.2 0.1 0 0 20 40 60 80 100 120 140 x 37 37/47

Smallest Eigenvalue as Well The cdf of the smallest eigenvalue, 38/47

Cdf s of smallest m5n4beta3.000.stag.a.least.fig eigenvalue 1 0.9 0.8 0.7 0.6 F(x) 0.5 0.4 0.3 0.2 0.1 0 0 2 4 6 8 10 12 14 16 x 39 39/47

Goals Continuum of Haar Measures generalizing orthogonal, unitary, symplectic New Definition of Jack Polynomials generalizing the zonals Computations! E.g. Moments of the Haar Measures Place finite random matrix theory β into same framework as infinite random matrix theory: specifically β as a knob to turn down the randomness, e.g. Airy Kernel d 2 /dx 2 +x+(2/β ½ )dw White Noise 40/47

Formally Let S n =2π/Γ(n/2)= surface area of sphere Defined at any n= β>0. A β ghost x is formally defined by a function f x (r) such that f x (r) r β 1 S β 1 dr=1. r=0 Note: For β integer, the x can be realized as a random spherically symmetric variable in β dimensions Example: A β normal ghost is defined by f(r)=(2π) β/2 e r2 /2 Example: Zero is defined with constant*δ(r). Can we do algebra? Can we do linear algebra? Can we add? Can we multiply? 41/47

A few more operations ΙΙxΙΙ is a real random variable whose density is given by f x (r) (x+x )/2 is real random variable given by multiplying ΙΙxΙΙ by a beta distributed random variable representing a coordinate on the sphere 42/47

Addition of Independent Ghosts: Addition returns a spherically symmetric object Have an integral formula Prefer: Add the real part, imaginary part completed to keep spherical symmetry 43/47

Multiplication of Independent Ghosts Just multiply ΙΙzΙΙ s and plug in spherical symmetry Multiplication is commutative (Important Example: Quaternions don t commute, but spherically symmetric random variables do!) 44/47

Understanding λ i λ j β Define volume element (dx)^ by (r dx)^=r β (dx)^ (β dim volume, like fractals, but don t really see any fractal theory here) Jacobians: A=QΛQ (Sym Eigendecomposition) Q daq=dλ+(q dq)λ Λ(Q dq) (da)^=(q daq)^= diagonal ^ strictly upper diagonal = dλ i =(dλ)^ off diag = ((Q dq) ij (λ i λ j ))^=(Q dq)^ λ i λ j β 45/47

Haar Measure β=1: E Q (trace(aqbq ) k )= C κ (A)C κ (B)/C κ (I) Forward Method: Suppose you know C κ s a priori. (Jack Polynomials!) Let A and B be diagonal indeterminants (Think Generating Functions) Then can formally obtain moments of Q: Example: E( q 11 2 q 22 2 ) = (n+α 1)/(n(n 1)(n+α)) α:=2/ β Can Gram Schmidt the ghosts. Same answers coming up! 46/47

Further Uses of Ghosts Multivariate Orthogonal Polynomials Tracy Widom Laws Largest Eigenvalues/Smallest Eigenvalues Expect lots of uses to be discovered 47/47