1. Advances in Random Matrix Theory (stochastic eigenanalysis)
Alan Edelman
MIT: Dept of Mathematics,
Computer Science AI Laboratories
11/23/2018

2. Stochastic Eigenanalysis
Counterpart to stochastic differential equations
Emphasis on applications to engineering & finance
Beautiful mathematics:
Random Matrix Theory
Free Probability
Raw Material from
Physics
Combinatorics
Numerical Linear Algebra
Multivariate Statistics
11/23/2018

3. Scalars, Vectors, Matrices
Mathematics: Notation = power & less ink!
Computation: Use those caches!
Statistics: Classical, Multivariate, 
Modern Random Matrix Theory
The Stochastic Eigenproblem
* Mathematics of probabilistic linear algebra
* Emerging Computational Algorithms
* Emerging Statistical Techniques
Ideas from numerical computation that stand the test of time are right for mathematics!

4. Open Questions Find new applications of spacing (or other) statistics
Cleanest derivation of Tracy-Widom?
“Finite” free probability?
Finite meets infinite
Muirhead meets Tracy-Widom
Software for stochastic eigen-analysis

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 Hermite Ensemble
Normalized eigenvalue histogram is a semi-circle
Precise statements require n etc.

6. 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 Hermite Ensemble
Normalized eigenvalue histogram is a semi-circle
Precise statements require n etc.
n x n iid standard normals

7. 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 Hermite Ensemble
Normalized eigenvalue histogram is a semi-circle
Precise statements require n etc.

8. Wigner’s original proof
Compute E(tr A2p) as n∞
Terms with too many indices, have some element with power 1. Vanishes with mean 0.
Terms with too few indices: not enough to be relevant as n∞
Leaves only a Catalan number left: Cp=(2p)/(p+1) for the moments when all is said and done
Semi-circle only distribution with Catalan number moments p

9. Finite Versions of semicircle
n=2; n=4;
n=3; n=5;

10. Finite Versions n=2; n=4; n=3; n=5;
Area under curve (-∞,x): Can be expressed as sums of probabilities that certain tridiagonal determinants are positive.

11. Wigner’s Semi-Circle Real Numbers: x β=1 Complex Numbers: x+iy β=2
Quaternions: x+iy+jz+kw β=4
β=2½? x+iy+jz β=2½?
Defined through joint eigenvalue density:
const x ∏|xi-xj|β ∏exp(-xi2 /2)
β=repulsion strength
β=0 “no interference” spacings are Poisson
Classical research only β=1,2,4 missing the link to Poisson, continuous techniques, etc

12. Largest eigenvalue
“convection-diffusion?”

13. Haar or not Haar? “Uniform Distribution on orthogonal matrices”
Gram-Schmidt or [Q,R]=QR(randn(n))

14. Haar or not Haar? “Uniform Distribution on orthogonal matrices”
Gram-Schmidt or [Q,R]=QR(randn(n)) 
Eigenvalues Wrong

15. Longest Increasing Subsequence (n=4) (Baik-Deift-Johansson) (Okounkov’s proof)
Green: 4 Yellow: 3 Red: 2 Purple: 1

16. Bulk spacing statistics
“convection-diffusion?”
Bus wait times in Mexico
Energy levels of heavy atoms
Parked Cars in London
Zeros of Riemann zeta
Mice Brain Wave Spikes
Telltale Sign: Repulsion + optimality

17. “what’s my β?” web page Cy’s tricks: Maximum Likelihood Estimation
Bayesian Probability
Kernel Density Estimation
Epanechnikov kernel
Confidence Intervals

18. Open Questions
Find new applications of spacing (or other) distributions
Cleanest derivation of Tracy-Widom?
“Finite” free probability?
Finite meets infinite
Muirhead meets Tracy-Widom
Software for stochastic eigen-analysis

19. Everyone’s Favorite Tridiagonal-2 1 1 n2 … … … … … d2
dx2
11/23/2018

20. Everyone’s Favorite Tridiagonal-2 1 G 1
(βn)1/2 1 n2 … … … + … … dW
β1/2 d2
dx2 +
11/23/2018

21. Stochastic Operator Limit, dW β 2 x dx d + - ,
N(0,2) χ n β 2 1 ~ H 2) (n 1) ÷ ø ö ç è æ - … … … , G β 2 H n +
Cast of characters: Dumitriu, Sutton, Rider

22. Open Questions
Find new applications of spacing (or other) distributions
Cleanest derivation of Tracy-Widom?
“Finite” free probability?
Finite meets infinite
Muirhead meets Tracy-Widom
Software for stochastic eigen-analysis

23. Is it really the random matrices?
The excitement is that the random matrix statistics are everyhwere
Random matrices properly tridiagonalized are discretizations of stochastic differential operators!
Eigenvalues of SDO’s not as well studied
Deep down this is what I believe is the important mechanism in the spacings, not the random matrices! (See Brian Sutton thesis, Brian Rider papers—connection to Schrodinger operators)
Deep down for other statistics, though it’s the matrices

24. Open Questions
Find new applications of spacing (or other) distributions
Cleanest derivation of Tracy-Widom?
“Finite” free probability?
Finite meets infinite
Muirhead meets Tracy-Widom
Software for stochastic eigen-analysis

25. Open Questions
Find new applications of spacing (or other) distributions
Cleanest derivation of Tracy-Widom?
“Finite” free probability?
Finite meets infinite
Muirhead meets Tracy-Widom
Software for stochastic eigen-analysis

26. Free Probability
Free Probability (name refers to “free algebras” meaning no strings attached)
Gets us past Gaussian ensembles and Wishart Matrices

27. The flipping coins example
Classical Probability: Coin: +1 or -1 with p=.5
50%
50%
50%
50% y: x:
x+y:

28. The flipping coins example
Classical Probability: Coin: +1 or -1 with p=.5
Free
50%
50%
50%
50%
eig(B):
eig(A):
eig(A+QBQ’):

29. From Finite to Infinite

30. From Finite to Infinite
 Gaussian (m=1)

31. From Finite to Infinite
 Gaussian (m=1)
Wiggly

32. From Finite to Infinite
 Gaussian (m=1)
Wiggly
Wigner

33. Semi-circle law for different betas

34. Open Questions
Find new applications of spacing (or other) distributions
Cleanest derivation of Tracy-Widom?
“Finite” free probability?
Finite meets infinite
Muirhead meets Tracy-Widom
Software for stochastic eigen-analysis

35. Matrix Statistics Many Worked out in 1950s and 1960s
Muirhead “Aspects of Multivariate Statistics”
Are two covariance matrices equal?
Does my matrix equal this matrix?
Is my matrix a multiple of the identity?
Answers Require Computation of
Hypergeometrics of Matrix Argument
Long thought Computationally Intractible

36. The special functions of multivariate statistics
Hypergeometric Functions of Matrix Argument
β=2: Schur Polynomials
Other values: Jack Polynomials
Orthogonal Polynomials of Matrix Argument
Begin with w(x) on I
∫ pκ(x)pλ(x) Δ(x)β ∏i w(xi)dxi = δκλ
Jack Polynomials orthogonal for w=1 on the unit circle. Analogs of xm
Plamen Koev revolutionary computation
Dumitriu’s MOPS symbolic package

37. ∫ pκ(x)pλ(x) Δ(x)β ∏i w(xi)dxi = δκλ
Multivariate Orthogonal Polynomials & Hypergeometrics of Matrix Argument
The important special functions of the 21st century
Begin with w(x) on I
∫ pκ(x)pλ(x) Δ(x)β ∏i w(xi)dxi = δκλ
Jack Polynomials orthogonal for w=1 on the unit circle. Analogs of xm

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

39. Multivariate Hypergeometric Functions

40. Multivariate Hypergeometric Functions

41. Open Questions
Find new applications of spacing (or other) distributions
Cleanest derivation of Tracy-Widom?
“Finite” free probability?
Finite meets infinite
Muirhead meets Tracy-Widom
Software for stochastic eigen-analysis

42. Plamen Koev’s clever idea

43. Symbolic MOPS applications
A=randn(n); S=(A+A’)/2; trace(S^4)
det(S^3)

44. Mops (Ioana Dumitriu) Symbolic

45. Random Matrix Calculator

46. Encoding the semicircle The algebraic secret
f(x) = sqrt(4-x2)/(2π)
m(z) = (-z + i*sqrt(4-z2))/2
L(m,z) ≡ m2+zm+1=0
m(z) = ∫ (x-z)-1f(x) dx Stieltjes transform
Practical encoding:
Polynomial L whose root m is Stieltjes transform

47. The Polynomial Method RMTool Eigenvectors as well!
The polynomial method for random matrices
Eigenvectors as well!

48. Plus + X =randn(n,n) A=X+X’ m2+zm+1=0 Y=randn(n,2n) B=Y*Y’
zm2+(2z-1)m+2=0
A+B
m3+(z+2)m2+(2z-1)m+2=0

49. Times * X =randn(n,n) A=X+X’ m2+zm+1=0 Y=randn(n,2n) B=Y*Y’
zm2+(2z-1)m+2=0
A*B
m4z2-2m3z+m2+4mz+4=0

50. Open Questions
Find new applications of spacing (or other) distributions
Cleanest derivation of Tracy-Widom?
“Finite” free probability?
Finite meets infinite
Muirhead meets Tracy-Widom
Software for stochastic eigen-analysis

51. Matrix Versions of Classical Stats
Orthog Matrix MATLAB (A=randn(n) B=randn(n))
Hermite
Sym Eig
eig(A+A’)
Normal
Laguerre
SVD
eig(A*A’)
Chi-squared
Jacobi
GSVD
gsvd(A,B)
Beta
Fourier
Eig
[U,R]=qr(A+i*B)

52. The big structure Hermite Sym Eig exp(-x2) Normal Laguerre SVD xαe-x
Orthog Matrix Weight Stats Graph Theory SymSpace
Hermite
Sym Eig
exp(-x2)
Normal
Complete Graph
non-compact
A,AI,AII
Laguerre
SVD
xαe-x
Chi-squared
Bipartite Graph
AIII,BDI,CII
Jacobi
GSVD
(1-x)α x
(1+x)β
Beta
Regular Graph
compact
A, AI, AII, C, D, CI, D, DIII
Fourier
Eig
eiθ
AIII, BDI, CDI

53. Summary
Stochastic Eigenanalysis
Emerging Techniques
Open Problems