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The Random Matrix Technique of Ghosts and Shadows

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Title: The Random Matrix Technique of Ghosts and Shadows


1
The Random Matrix Technique of Ghosts and Shadows
  • Alan Edelman
  • Dept of Mathematics
  • Computer Science and AI Laboratories
  • Massachusetts Institute of Technology
  • (with thanks to Plamen Koev)

2
Short followed by the Movie
  • Some interesting computational techniques
  • Random Matrix Theory Theorems, Applications,
    and Software
  • A new application can be more valuable than a
    theorem!
  • A well crafted experiment or package is not a
    theorem but it can be as important or even more
    to the field!
  • The Main Show The Method of Ghosts and Shadows
    in Random Matrix Theory
  • Yet another nail in the threefold way coffin

3
Semi-Circle Law
  • Naïve Way
  • MATLAB Arandn(n) S(AA)/sqrt(2n)eig
    (S)
  • R
  • Amatrix(rnorm(nn),ncoln)S(at(a))/sqrt(
    2n)eigen(S,symmetricT,only.valuesT)values
  • Mathematica ARandomArrayNormalDistribution,n
    ,nS(ATransposeA)/SqrtnEigenvaluess

4
Compute All the Eigenvalues
  • Sym Tridiagonal ß1real, ß2complex,
    ß4quaternion, ß2½?

Diagonals N(0,2), Off-diagonals chi
random-variables N2000 12 seconds vs. 0.2
seconds (factor of 60!!) (Dumitriu E 2002)
5
Histogram without HistogrammingSturm Sequences
  • Count eigs lt 0.5 Count sign changes in
  • Det (A-0.5I)1k,1k
  • Count eigs in x,xh
  • Take difference in number of sign changes at
    xh and x

Mentioned in Dumitriu and E 2006, Used
theoretically in Albrecht, Chan, and E 2008
6
A good computational trick is a good theoretical
trick!
Finite Semi-Circle Laws for Any Beta!
Finite Tracy-Widom Laws for Any Beta!
7
Stochastic Differential Eigen-Equations
  • Tridiagonal Models Suggest SDEs 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 todays talk)

8
Tracy-Widom ComputationsEigenvalues without
the whole matrix!
Never construct the entire tridiagonal
matrix! Just say the upper 10n1/3 by 10n1/3
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
9
Other Computational Results
  • Infinite Random Matrix Theory
  • The Free Probability Calculator (Raj)
  • Finite Random Matrix Theory
  • MOPS (Ioana Dumitriu) (ß orthogonal polynomials)
  • Hypergeometrics of Matrix Argument (Plamen Koev)
    (ß distribution functions for finite stats such
    as the finite Tracy-Widom laws for Laguerre)
  • Good stuff, but not today.

10
Ghosts and Shadows
11
Scary Ideas in Mathematics
  • Zero
  • Negative
  • Radical
  • Irrational
  • Imaginary
  • Ghosts Something like a commutative algebra of
    random variables that generalizes Reals,
    Complexes, and Quaternions and inspires
    theoretical results and numerical computation

12
RMT Densities
  • Hermite
  • c ??i-?jß e-??i2/2 (Gaussian Ensemble)
  • Laguerre
  • c ??i-?jß ??im e-??i (Wishart Matrices)
  • Jacobi
  • c ??i-?jß ??im1 ?(1-?i)m2 (Manova Matrices)
  • Fourier
  • c ??i-?jß (on the complex unit circle) Jack
    Polynomials

Traditional Story Count the real parameters
ß1,2,4 for real, complex, quaternion Application
s ß1 All of Multivariate Statistics ß2parked
cars in London, wireless networks
ß4 There and almost nobody cares Dyson 1962
Threefold Way Three Division Rings
13
ß-Ghosts
  • ß1 has one real Gaussian (G)
  • ß2 has two real Gaussians (GiG)
  • ß4 has four real Gaussians (GiGjGkG)
  • ß1 has one real part and (ß-1) Ghost parts

14
Introductory Theory
  • There is an advanced theory emerging (some other
    day)
  • Informally
  • A ß-ghost is a spherically symmetric random
    variable defined on Rß
  • A shadow is a derived real or complex quantity

15
Goals
  • Continuum of Haar Measureas 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 ßinto same
    framework as infinite random matrix theory
    specifically ß as a knob to turn down the
    randomness, e.g. Airy Kernel
  • d2/dx2x(2/ß½)dW ?White Noise

16
Formally
  • Let Sn2p/G(n/2)suface area of sphere
  • Defined at any n ßgt0.
  • A ß-ghost x is formally defined by a function
    fx(r) such that ?8 fx(r) rß-1Sß-1dr1.
  • Note For ß integer, the x can be realized as a
    random spherically symmetric variable in ß
    dimensions
  • Example A ß-normal ghost is defined by
    f(r)(2p)-ß/2e-r2/2
  • Example Zero is defined with constantd(r).
  • Can we do algebra? Can we do linear algebra?
  • Can we add? Can we multiply?

r0
17
A few more operations
  • ??x?? is a real random variable whose density is
    given by fx(r)
  • (xx)/2 is real random variable given by
    multiplying ??x?? by a beta distributed random
    variable representing a coordinate on the sphere

18
Representations
  • Ive tried a few on for size. My favorite right
    now is
  • A complex number z with ??z??, the radius and
    Re(z), the real part.

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

19
Multiplication of Independent Ghosts
  • Just multiply ??z??s and plug in spherical
    symmetry
  • Multiplication is commutative
  • (Important Example Quaternions dont commute,
    but spherically symmetric random variables do!)

20
Shadow Example
  • Given a ghost Gaussian, Gß, the length is a real
    chi-beta variable ? ß variable.

21
Linear Algebra Example
  • Given a ghost Gaussian, Gß, the length is a real
    chi-beta variable ? ß variable.
  • Gram-Schmidt
  • ? ? ?
  • or
  • Q

Gß Gß Gß
Gß Gß Gß
Gß Gß Gß
?3ß Gß Gß
Gß Gß
Gß Gß
?3ß Gß Gß
?2ß Gß

?3ß Gß Gß
?2ß Gß

H3
H2
H1
Gß Gß Gß
Gß Gß Gß
Gß Gß Gß
?3ß Gß Gß
?2ß Gß

Q has ß -Haar Measure! We have computed
moments! (more later)
22
Tridiagonalizing Example
Gß Gß Gß Gß
Gß Gß Gß Gß
Gß Gß Gß Gß

Gß Gß Gß Gß Gß
Symmetric Part of
?
23
Understanding ??i-?jß
  • Define volume element (dx) by
  • (r dx)rß(dx) (ß-dim volume, like fractals,
    but dont really see any fractal theory here)
  • Jacobians AQ?Q (Sym Eigendecomposition)
  • QdAQd?(QdQ)?- ?(QdQ)
  • (dA)(QdAQ) diagonal strictly-upper
  • diagonal ?d?i (d?)
  • off-diag ?((QdQ)ij(?i-?j))(QdQ)
    ??i-?jß

24
Haar Measure
  • ß1 EQ(trace(AQBQ)k)?C?(A)C?(B)/C?(I)
  • Forward Method Suppose you know C?s a-priori.
    (Jack Polynomials!)
  • Let A and B be diagonal indeterminants (Think
    Generating Functions)
  • Then can formally obtain moments of Q
  • Example E(q112q222) (na-1)/(n(n-1)(na))
  • a2/ ß
  • Can Gram-Schmidt the ghosts. Same answers coming
    up!

25
Further Uses of Ghosts
  • Multivariate Othogonal Polynomials
  • Largest Eigenvalues/Smallest Eigenvalues
  • Noncentral Distributions
  • Expect Lots of Uses to be discovered
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