Exponential tail inequalities for eigenvalues of random matrices
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1 Exponential tail inequalities for eigenvalues of random matrices M. Ledoux Institut de Mathématiques de Toulouse, France
2 exponential tail inequalities classical theme in probability and statistics quantify the asymptotic statements central limit theorems large deviation principles
3 classical exponential inequalities sum of independent random variables S n = 1 n (X X n ) 0 X i 1 independent P ( S n E(S n ) + t ) e t2 /2, t 0 Hoeffding s inequality same as for X i standard Gaussian central limit theorem
4 measure concentration ideas asymptotic geometric analysis V. D. Milman (1970) S n = 1 n (X X n ) F (X ) = F (X 1,..., X n ), F : R n R Lipschitz Gaussian sample independent random variables (M. Talagrand 1995)
5 concentration inequalities S n = 1 n (X X n ) F (X ) = F (X 1,..., X n ), F : R n R 1-Lipschitz X 1,..., X n independenty standard Gaussian ( P F (X ) E ( F (X ) ) ) + t e t2 /2, t 0 0 X i 1 independent, F 1-Lipschitz and convex ( P F (X ) E ( F (X ) ) ) + t 2 e t2 /4, t 0 M. Talagrand (1995)
6 concentration inequalities numerous applications geometric functional analysis discrete and combinatorial probability empirical processes statistical mechanics random matrix theory
7 recent studies of random matrix and random growth models new asymptotics common, non central, rate (mean) 1/3 universal limiting Tracy-Widom distribution random matrices, longest increasing subsequence, random growth models, last passage percolation...
8 random matrix models Wigner matrix X N = (X N ij ) 1 i,j N symmetric N N matrix X N ij, i j, independent identically distributed E(X N ij ) = 0, E( (X N ij )2) = 1 eigenvalues λ N 1 λn N of X N asymptotics of the eigenvalues as the size N scaling X N = X N / N
9 Wigner s theorem (1955) asymptotic behavior of the spectral measure ( λ N k = λn k / N) 1 N N dν(x) = 1 δ λn 4 x k 2π 2 dx on ( 2, +2)
10 1 N N Wigner s theorem dν(x) = 1 δ λn 4 x k 2π 2 dx semi-circle law global regime large deviation asymptotics of the spectral measure fluctuations of the spectral measure N [ ) ] f ( λn k f dν G Gaussian variable f : R R smooth
11 1 N N Wigner s theorem dν(x) = 1 δ λn 4 x k 2π 2 dx semi-circle law local regime behavior of the individual eigenvalues spacings (bulk behavior) extremal eigenvalues (edge behavior)
12 extremal eigenvalues largest eigenvalue λ N N = max 1 k N λ N k λ N N = λn N / N
13 1 N N Wigner s theorem dν(x) = 1 δ λn 4 x k 2π 2 dx on ( 2, +2) local regime behavior of the individual eigenvalues spacings (bulk behavior) extremal eigenvalues (edge behavior)
14 extremal eigenvalues largest eigenvalue λ N N = max 1 k N λ N k λ N N = λn N / N 2 fluctuations around 2 Gaussian Orthogonal or Unitary Ensemble (GOE, GUE) N 2/3[ λn N 2 ] F TW C. Tracy, H. Widom (1994) rate (mean) 1/3 : N 1/6[ λ N N 2 N ] GUE ( F TW (s) = exp s (x s)u(x) 2 dx ), s R u = 2u 3 + xu Painlevé II equation (similar for GOE)
15 Gaussian Orthogonal or Unitary Ensemble (GOE, GUE) independent entries X N ij real or complex Gaussian P(dX ) = 1 Z exp ( Tr ( βx 2 /4 )) dx dx Lebesgue measure on N N matrices orthogonal (β = 1) or unitary conjugation (β = 2) joint law of the eigenvalues (λ N 1 λn N ) of X N
16 joint law of the eigenvalues (λ N 1 λn N ) of X N P N (dx) = 1 Z N (x) β N x = (x 1,..., x N ) R N e βx2 k /4 dx k N (x) = k<l(x k x l ) Vandermonde determinant β = 1 : real β = 2 : complex large deviation principles (Laplace methods) fluctuations of the spectral measure
17 law of (λ N 1,..., λ N N ) : PN (dx) = 1 Z N (x) β fluctuations of extremal eigenvalues N e βx2 k /4 dx k N 2/3[ λn N 2 ] F TW Tracy-Widom distribution orthogonal polynomial method (β = 2) marginals of (λ N 1,..., λn N ) in terms of the kernel K N (x, y) = N 1 l=0 P l (x)p l (y) P l Hermite orthogonal polynomials for dµ(x) = e βx2 /4 dx Z orthogonal polynomial asymptotics
18 fluctuations of extremal eigenvalues recent years : extensions to (non Gaussian) Wigner matrices comparison with the Gaussian case N 2/3[ λn N 2 ] F TW Tracy-Widom distribution A. Soshnikov (1999) combinatorial moment method ( N ) ( ) E λ N p k ( = E Tr ( (X N ) p ) ), p = O(N 2/3 ) T. Tao, V. Vu (2009) Lindeberg method
19 survey of recent approaches to non asymptotic exponential inequalities quantify the limit theorems spectral measure extremal eigenvalues catch the new rate (mean) 1/3
20 two main questions and objectives tail inequalities for the spectral measure ( N ) P f ( λ N k ) t
21 1 N N Wigner s theorem dν(x) = 1 δ λn 4 x k 2π 2 dx semi-circle law global regime large deviation asymptotics of the spectral measure fluctuations of the spectral measure N [ ) ] f ( λn k f dν G Gaussian variable f : R R smooth
22 two main questions and objectives tail inequalities for the spectral measure ( N ) P f ( λ N k ) t tail inequalities for the extremal eigenvalues P ( λn N 2 + ε ), P ( λn N 2 ε )
23 extremal eigenvalues largest eigenvalue λ N N = max 1 k N λ N k λ N N = λn N / N 2 fluctuations around 2 Gaussian Orthogonal or Unitary Ensemble (GOE, GUE) N 2/3[ λn N 2 ] F TW Tracy-Widom distribution ( F TW (s) = exp u = 2u 3 + xu s (x s)u(x) 2 dx Painlevé II equation ), s R
24 two main questions and objectives tail inequalities for the spectral measure ( N ) P f ( λ N k ) t tail inequalities for the extremal eigenvalues P ( λn N 2 + ε ), P ( λn N 2 ε ) Gaussian and more general Wigner matrices measure concentration tool : F = F (X N ) = F (X N ij )
25 two main questions and objectives tail inequalities for the spectral measure ( N ) P f ( λ N k ) t tail inequalities for the extremal eigenvalues P ( λn N 2 + ε ), P ( λn N 2 ε ) Gaussian and more general Wigner matrices measure concentration tool : F = F (X N ) = F (X N ij )
26 tail inequalities for the spectral measure A. Guionnet, O. Zeitouni (2000) f smooth measure concentration f : R R Lipschitz F : X N Tr f ( N X ) N = f ( λn ) k with respect to the Euclidean structure on Lipschitz N N matrices convex if f is convex
27 concentration inequalities S n = 1 n (X X n ) F (X ) = F (X 1,..., X n ), F : R n R 1-Lipschitz X 1,..., X n independenty standard Gaussian ( P F (X ) E ( F (X ) ) ) + t e t2 /2, t 0 0 X i 1 independent, F 1-Lipschitz and convex ( P F (X ) E ( F (X ) ) ) + t 2 e t2 /4, t 0 M. Talagrand (1995)
28 tail inequalities for the spectral measure A. Guionnet, O. Zeitouni (2000) f smooth measure concentration f : R R 1-Lipschitz F : X N Tr f ( N X ) N = f ( λn ) k with respect to the Euclidean structure on Lipschitz N N matrices convex if f is convex ( N [ P f ( λ N k ) E( f ( λ N i ) )] ) t C e t2 /C, t 0
29 1 N N Wigner s theorem (1955) dν(x) = 1 δ λn 4 x k 2π 2 dx semi-circle law global regime large deviation asymptotics of the spectral measure fluctuations of the spectral measure N [ ) ] f ( λn k f dν G Gaussian variable f : R R smooth
30 tail inequalities for the spectral measure A. Guionnet, O. Zeitouni (2000) f smooth measure concentration f : R R 1-Lipschitz X N Tr f ( X N ) = N f ( λn ) k with respect to the Euclidean structure on convex if f is convex Lipschitz N N matrices ( N [ P f ( λ N k ) E( f ( λ N i ) )] ) t C e t2 /C, t 0
31 non Lipschitz functions f typically f = 1 A, A interval N f ( λn ) k = # { λn k A } counting function suitable X N = (Xij N ), A interval in ( 2, +2) 1 log N N [ f ( λn k ) E ( f ( λn k ) )] G Gaussian variable exponential tail inequalities (GUE) ( N [ ) P f ( λ N k ) E( f ( λ N k ))] t C e ct log(1+t/ log N), t 0 non Gaussian Wigner matrices?
32 partial results (Stieltjes transform) control of the Kolmogorov distance in Wigner s theorem F. Götze, A. Tikhomirov crucial tool in the recent solution (2009) of the (bulk) spacings for Wigner matrices L. Erdös, S. Péché, J. Ramirez, B. Schlein, H.-T. Yau V. Vu, T. Tao ( P # { λn k A } ) t A N C e t A N/C, t C Wigner s law at small scales A 1 N
33 two main questions and objectives tail inequalities for the spectral measure ( N ) P f ( λ N k ) t tail inequalities for the extremal eigenvalues P ( λn N 2 + ε ), P ( λn N 2 ε ) Gaussian and more general Wigner matrices measure concentration tool : F = F (X N ) = F (X N ij )
34 two main questions and objectives tail inequalities for the spectral measure ( N ) P f ( λ N k ) t tail inequalities for the extremal eigenvalues P ( λn N 2 + ε ), P ( λn N 2 ε ) Gaussian and more general Wigner matrices measure concentration tool : F = F (X N ) = F (X N ij )
35 tail inequalities for the extremal eigenvalues fluctuations of the largest eigenvalue N 2/3[ λn N 2 ] F TW Tracy-Widom distribution P ( λn N 2 + s N 2/3) F TW (s), s R finite N inequalities at the (mean) 1/3 rate reflecting the tails of F TW bounds on Var( λ N N )
36 N 2/3[ λn N 2 ] F TW GUE ( F TW (s) = exp u = 2u 3 + xu Tracy-Widom distribution s (x s)u(x) 2 dx Painlevé II equation ), s R density
37 mean 1.77 F TW (s) e s3 /12 as s 1 F TW (s) e 4s3/2 /3 as s + density (similar for GOE)
38 non asymptotic small deviation inequalities P ( λn N 2 + s N 2/3) F TW (s), s R expected bounds P ( λn N 2 + ε )
39 non asymptotic small deviation inequalities P ( λn N 2 + s N 2/3) F TW (s), s R 1 F TW (s) e s3/2 /C (s + ) expected bounds P ( λn N 2 + ε ) C e N ε3/2 /C, 0 < ε 1 fit the Tracy-Widom asymptotics (ε = s N 2/3 )
40 non asymptotic small deviation inequalities P ( λn N 2 + s N 2/3) F TW (s), s R 1 F TW (s) e s3/2 /C (s + ) expected bounds P ( λn N 2 + ε ) C e N ε3/2 /C, 0 < ε 1 P ( λn N 2 ε ) fit the Tracy-Widom asymptotics (ε = s N 2/3 )
41 non asymptotic small deviation inequalities P ( λn N 2 + s N 2/3) F TW (s), s R 1 F TW (s) e s3/2 /C (s + ) F TW (s) e s3 /C (s ) expected bounds P ( λn N 2 + ε ) C e N ε3/2 /C, 0 < ε 1 P ( λn N 2 ε ) fit the Tracy-Widom asymptotics (ε = s N 2/3 )
42 non asymptotic small deviation inequalities P ( λn N 2 + s N 2/3) F TW (s), s R 1 F TW (s) e s3/2 /C (s + ) F TW (s) e s3 /C (s ) expected bounds P ( λn N 2 + ε ) C e N ε3/2 /C, 0 < ε 1 P ( λn N 2 ε ) C e N2 ε 3 /C, 0 < ε 1 fit the Tracy-Widom asymptotics (ε = s N 2/3 )
43 general concentration inequalities useless Gaussian matrix λ N N = sup X N v, v v =1 λ N N Lipschitz of the Gaussian entries X N ij Gaussian concentration P ( λn N E( λ N N ) + t) e N t2 /C, t 0 E( λ N N ) 2 does not fit the (small deviation) regime t = s N 2/3 correct large deviation bounds (t 1)
44 several (specific) approaches variety of techniques to the non asymptotic inequalities fragmentary results : geometric percolation model orthogonal polynomial bounds, Riemann-Hilbert methods (GUE, GOE) moment recurrence equations (GUE, GOE, invariant ensembles) combinatorial moment method (Wigner, sample covariance matrices) tridiagonal matrices (β-ensembles)
45 moment recurrence equations Gaussian Unitary Ensemble (GUE) Harer-Zagier (1986) (map enumeration) ( ap N = E Tr ( (X N ) 2p ) ) ( N = E (λ N k ), )2p p N three term recurrence equation (p + 1)a N p = (4p 2)Na N p 1 + (p 1)(2p 1)(2p 3)aN p 2 ( ap N = E Tr ( (X N ) 2p ) ) C (4N) p e Cp3 /N 2, p 3 N 2 P ( λn N 2 + ε ) C e N ε3/2 /C, 0 < ε 1
46 extends to various unitary invariant models P(dX ) = 1 Z exp ( Tr ( v(x ) )) dx, v : R R dµ = e v dx Z classical orthogonal polynomials continuous : Hermite, Laguerre, Jacobi discrete : Charlier, Meixner, Krawtchouk, Hahn oriented percolation model length of the longest increasing subsequence real Gaussian Orthogonal Ensemble (GOE) five term recurrence equation
47 combinatorial moment method Wigner matrices (proof of Wigner s theorem) ( N ) E (λ N k )p = E (Tr ( (X N ) p ) ) asymptotic results : A. Soshnikov (1999) O. Feldheim, S. Sodin (2009) non asymptotic moment inequalities P ( λn N 2 + ε ) C e N ε3/2 /C, 0 < ε 1 entries X N ij symmetric, subgaussian extension to sample covariance matrices
48 sample covariance matrices multivariate statistical inference principal component analysis population (Y 1,..., Y N ) Y j vector in R M (characters) sample covariance matrix Y Y M κn, N (Gaussian) Wishart matrix models
49 sample covariance matrices Y = Y M,N = (Y ij ) 1 i M,1 j N M N matrix (M N) Y ij independent identically distributed E(Y ij ) = 0, E ( (Y ij ) 2) = 1 eigenvalues 0 λ N 1 λn N of Y Y asymptotic spectral measure M κ N, κ 1 1 N λ N k = λn k /N N ρ on ( ( κ 1) 2, ( κ + 1) 2) δ λn k ρ Marchenko-Pastur distribution
50 Tracy-Widom theorem for the largest eigenvalue O. Feldheim, S. Sodin (2009) non asymptotic moment inequalities largest eigenvalue (symmetric, subgaussian entries) P ( λn N ( κ + 1) 2 + ε ) C e N ε3/2 /C, 0 < ε 1
51 sample covariance matrices Y = Y M,N = (Y ij ) 1 i M,1 j N M N matrix (M N) Y ij independent identically distributed E(Y ij ) = 0, E ( (Y ij ) 2) = 1 eigenvalues 0 λ N 1 λn N of Y Y asymptotic spectral measure M κ N, κ 1 1 N λ N k = λn k /N N ρ on ( ( κ 1) 2, ( κ + 1) 2) δ λn k ρ Marchenko-Pastur distribution
52 Tracy-Widom theorem for the largest eigenvalue O. Feldheim and S. Sodin (2009) non asymptotic moment inequalities largest eigenvalue (symmetric, subgaussian entries) P ( λn N ( κ + 1) 2 + ε ) C e N ε3/2 /C, 0 < ε 1 smallest eigenvalue at the soft edge M κ N, κ > 1 P ( λn 1 ( κ 1) 2 ε ) C e N ε3/2 /C, 0 < ε 1
53 tridiagonal models (β-ensembles) GOE, GUE, Wishart models joint law of the eigenvalues (λ N 1 λn N ) of X N P N (dx) = 1 Z N (x) β N e βx2 k /4 dx k β = 1 : GOE β = 2 : GUE
54 tridiagonal representation g 1 χ N χ N 1 g 2 χ N χ N 2 g 3 χ.. N χ2 g N 1 χ χ 1 g N g 1,..., g N independent standard normal χ N 1,..., χ 1 independent chi-variables same GOE eigenvalues H. Trotter (1984), A. Edelman, I. Dimitriu (2002)
55 tridiagonal models (β-ensembles) GOE, GUE, Wishart models joint law of the eigenvalues (λ N 1 λn N ) of X N P N (dx) = 1 Z N (x) β N e βx2 k /4 dx k β = 1 : GOE β = 2 : GUE
56 tridiagonal representation g 1 χ (N 1)β 0 0 χ (N 1)β g 2 χ (N 2)β χ (N 2)β g 3 χ.. (N 3)β χ2β g N 1 χ β 0 0 χ β g N g 1,..., g N independent standard normal χ (N 1)β,..., χ β independent chi-variables same GOE eigenvalues H. Trotter (1984), A. Edelman, I. Dimitriu (2002) similar for the β-ensembles, Wishart models
57 A. Ramirez, B. Rider, B. Virag (2007) new proof of Tracy-Widom theorem (every β) new description of the Tracy-Widom distribution { 2 TW β = sup β f 0 f 2 (x) db(x) f (0) = 0, 0 0 f 2 (x)dx = 1 0 [f (x) 2 + x f 2 (x)]dx < [ f (x) 2 + x f 2 (x) ] } dx Painlevé equation?
58 bounds on the largest eigenvalue λ N N = max 1 k N λ N k
59 tridiagonal representation g 1 χ N χ N 1 g 2 χ N χ N 2 g 3 χ.. N χ2 g N 1 χ χ 1 g N g 1,..., g N independent standard normal χ N 1,..., χ 1 independent chi-variables
60 bounds on the largest eigenvalue λ N N = max 1 k N λ N k ( λ N N = sup N g i vi v =1 i=1 N 1 i=1 χ N i v i v i+1 ) explicit computations P ( λn N 2 + ε ) C e N ε3/2 /C, 0 < ε 1 P ( λn N 2 ε ) C e N2 ε 3 /C, 0 < ε 1 β-ensembles (GOE, GUE, Wishart models)
61 bounds on the largest eigenvalue λ N N = max 1 k N λ N k ( λ N N = sup N g i vi v =1 i=1 N 1 i=1 χ N i v i v i+1 ) explicit computations P ( λn N 2 + ε ) 1 C e C N ε3/2, 0 < ε 1 P ( λn N 2 ε ) 1 C e C N2 ε 3, 0 < ε 1 β-ensembles (GOE, GUE, Wishart models)
62 P ( λn N 2 + ε ) C e N ε3/2 /C, 0 < ε 1 P ( λn N 2 ε ) C e N2 ε 3 /C, 0 < ε 1 finite N variance bounds Var( λ N N ) C N 4/3 N 2/3[ λn N 2 ] F TW Tracy-Widom distribution β-ensembles (GOE, GUE, Wishart models) open for general Wigner matrices
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