Concentration Inequalities for Random Matrices M. Ledoux Institut - - PowerPoint PPT Presentation

Concentration Inequalities for Random Matrices

• M. Ledoux

Institut de Math´ ematiques de Toulouse, France

exponential tail inequalities classical theme in probability and statistics

exponential tail inequalities classical theme in probability and statistics quantify the asymptotic statements

exponential tail inequalities classical theme in probability and statistics quantify the asymptotic statements central limit theorems large deviation principles

classical exponential inequalities sum of independent random variables Sn = 1 √n (X1 + · · · + Xn)

classical exponential inequalities sum of independent random variables Sn = 1 √n (X1 + · · · + Xn) 0 ≤ Xi ≤ 1 independent P

• Sn ≥ E(Sn) + t
• ≤ e−t2/2,

t ≥ 0 Hoeffding’s inequality

classical exponential inequalities sum of independent random variables Sn = 1 √n (X1 + · · · + Xn) 0 ≤ Xi ≤ 1 independent P

• Sn ≥ E(Sn) + t
• ≤ e−t2/2,

t ≥ 0 Hoeffding’s inequality same as for Xi standard Gaussian central limit theorem

measure concentration ideas

measure concentration ideas asymptotic geometric analysis

• V. Milman (1970)

measure concentration ideas asymptotic geometric analysis

• V. Milman (1970)

Sn = 1 √n (X1 + · · · + Xn) F(X) = F(X1, . . . , Xn), F : Rn → R Lipschitz

measure concentration ideas asymptotic geometric analysis

• V. Milman (1970)

Sn = 1 √n (X1 + · · · + Xn) F(X) = F(X1, . . . , Xn), F : Rn → R Lipschitz Gaussian sample

measure concentration ideas asymptotic geometric analysis

• V. Milman (1970)

Sn = 1 √n (X1 + · · · + Xn) F(X) = F(X1, . . . , Xn), F : Rn → R Lipschitz Gaussian sample independent random variables

concentration inequalities Sn = 1 √n (X1 + · · · + Xn) F(X) = F(X1, . . . , Xn), F : Rn → R 1-Lipschitz

concentration inequalities Sn = 1 √n (X1 + · · · + Xn) F(X) = F(X1, . . . , Xn), F : Rn → R 1-Lipschitz X1, . . . , Xn independenty standard Gaussian P

• F(X) ≥ E
• F(X)
• + t
• ≤ e−t2/2,

t ≥ 0

concentration inequalities Sn = 1 √n (X1 + · · · + Xn) F(X) = F(X1, . . . , Xn), F : Rn → R 1-Lipschitz X1, . . . , Xn independenty standard Gaussian P

• F(X) ≥ E
• F(X)
• + t
• ≤ e−t2/2,

t ≥ 0 0 ≤ Xi ≤ 1 independent, F 1-Lipschitz

concentration inequalities Sn = 1 √n (X1 + · · · + Xn) F(X) = F(X1, . . . , Xn), F : Rn → R 1-Lipschitz X1, . . . , Xn independenty standard Gaussian P

• F(X) ≥ E
• F(X)
• + t
• ≤ e−t2/2,

t ≥ 0 0 ≤ Xi ≤ 1 independent, F 1-Lipschitz and convex

concentration inequalities Sn = 1 √n (X1 + · · · + Xn) F(X) = F(X1, . . . , Xn), F : Rn → R 1-Lipschitz X1, . . . , Xn independenty standard Gaussian P

• F(X) ≥ E
• F(X)
• + t
• ≤ e−t2/2,

t ≥ 0 0 ≤ Xi ≤ 1 independent, F 1-Lipschitz and convex P

• F(X) ≥ E
• F(X)
• + t
• ≤ 2 e−t2/4,

t ≥ 0

concentration inequalities Sn = 1 √n (X1 + · · · + Xn) F(X) = F(X1, . . . , Xn), F : Rn → R 1-Lipschitz X1, . . . , Xn independenty standard Gaussian P

• F(X) ≥ E
• F(X)
• + t
• ≤ e−t2/2,

t ≥ 0 0 ≤ Xi ≤ 1 independent, F 1-Lipschitz and convex P

• F(X) ≥ E
• F(X)
• + t
• ≤ 2 e−t2/4,

t ≥ 0

• M. Talagrand (1995)

empirical processes X1, . . . , Xn independent with values in (S, S) F collection of functions f : S → [0, 1]

empirical processes X1, . . . , Xn independent with values in (S, S) F collection of functions f : S → [0, 1] Z = sup

f ∈F n

• i=1

f (Xi)

empirical processes X1, . . . , Xn independent with values in (S, S) F collection of functions f : S → [0, 1] Z = sup

f ∈F n

• i=1

f (Xi) Z Lipschitz and convex

empirical processes X1, . . . , Xn independent with values in (S, S) F collection of functions f : S → [0, 1] Z = sup

f ∈F n

• i=1

f (Xi) Z Lipschitz and convex concentration inequalities on P

• Z − E(Z)
• ≥ t
• ,

t ≥ 0

Z = sup

f ∈F n

• i=1

f (Xi) |f | ≤ 1, E(f (Xi)) = 0, f ∈ F

Z = sup

f ∈F n

• i=1

f (Xi) |f | ≤ 1, E(f (Xi)) = 0, f ∈ F P

• |Z − M| ≥ t
• ≤ C exp
• − t

C log

• 1 +

t σ2 + M

• ,

t ≥ 0 C > 0 numerical constant, M mean or median of Z σ2 = supf ∈F n

i=1 E(f 2(Xi))

Z = sup

f ∈F n

• i=1

f (Xi) |f | ≤ 1, E(f (Xi)) = 0, f ∈ F P

• |Z − M| ≥ t
• ≤ C exp
• − t

C log

• 1 +

t σ2 + M

• ,

t ≥ 0 C > 0 numerical constant, M mean or median of Z σ2 = supf ∈F n

i=1 E(f 2(Xi))

• M. Talagrand (1996)

Z = sup

f ∈F n

• i=1

f (Xi) |f | ≤ 1, E(f (Xi)) = 0, f ∈ F P

• |Z − M| ≥ t
• ≤ C exp
• − t

C log

• 1 +

t σ2 + M

• ,

t ≥ 0 C > 0 numerical constant, M mean or median of Z σ2 = supf ∈F n

i=1 E(f 2(Xi))

• M. Talagrand (1996)
• P. Massart (2000)
• S. Boucheron, G. Lugosi, P. Massart (2005)

Z = sup

f ∈F n

• i=1

f (Xi) |f | ≤ 1, E(f (Xi)) = 0, f ∈ F P

• |Z − M| ≥ t
• ≤ C exp
• − t

C log

• 1 +

t σ2 + M

• ,

t ≥ 0 C > 0 numerical constant, M mean or median of Z σ2 = supf ∈F n

i=1 E(f 2(Xi))

• M. Talagrand (1996)
• P. Massart (2000)
• S. Boucheron, G. Lugosi, P. Massart (2005)

P.-M. Samson (2000) (dependence)

concentration inequalities numerous applications

concentration inequalities numerous applications

• geometric functional analysis
• discrete and combinatorial probability
• empirical processes
• statistical mechanics
• random matrix theory

concentration inequalities numerous applications

• geometric functional analysis
• discrete and combinatorial probability
• empirical processes
• statistical mechanics
• random matrix theory

recent studies of random matrix and random growth models

recent studies of random matrix and random growth models new asymptotics

recent studies of random matrix and random growth models new asymptotics common, non-central, rate (mean)1/3

recent studies of random matrix and random growth models new asymptotics common, non-central, rate (mean)1/3 universal limiting Tracy-Widom distribution

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...

sample covariance matrices multivariate statistical inference principal component analysis

sample covariance matrices multivariate statistical inference principal component analysis population (Y1, . . . , YN) Yj vectors (column) in RM (characters)

sample covariance matrices multivariate statistical inference principal component analysis population (Y1, . . . , YN) Yj vectors (column) in RM (characters) Y = (Y1, . . . , YN) M × N matrix

sample covariance matrices multivariate statistical inference principal component analysis population (Y1, . . . , YN) Yj vectors (column) in RM (characters) Y = (Y1, . . . , YN) M × N matrix sample covariance matrix Y Y t (M × M)

sample covariance matrices multivariate statistical inference principal component analysis population (Y1, . . . , YN) Yj vectors (column) in RM (characters) Y = (Y1, . . . , YN) M × N matrix sample covariance matrix Y Y t (M × M) (independent) Gaussian Yj : Wishart matrix models

is Y Y t a good approximation of the population covariance matrix E(Y Y t) ?

is Y Y t a good approximation of the population covariance matrix E(Y Y t) ? M finite 1 N Y Y t → E(Y Y t) N → ∞

is Y Y t a good approximation of the population covariance matrix E(Y Y t) ? M finite 1 N Y Y t → E(Y Y t) N → ∞ M infinite ?

is Y Y t a good approximation of the population covariance matrix E(Y Y t) ? M finite 1 N Y Y t → E(Y Y t) N → ∞ M infinite ? M = M(N) → ∞ N → ∞

is Y Y t a good approximation of the population covariance matrix E(Y Y t) ? M finite 1 N Y Y t → E(Y Y t) N → ∞ M infinite ? M = M(N) → ∞ N → ∞ M N ∼ ρ ∈ (0, ∞) N → ∞

sample covariance matrices Y = (Y1, . . . , YN) M × N matrix

sample covariance matrices Y = (Y1, . . . , YN) M × N matrix Y = (Yij)1≤i≤M,1≤j≤N Yij independent identically distributed (real or complex) E(Yij) = 0, E(Y 2

ij ) = 1

sample covariance matrices Y = (Y1, . . . , YN) M × N matrix Y = (Yij)1≤i≤M,1≤j≤N Yij independent identically distributed (real or complex) E(Yij) = 0, E(Y 2

ij ) = 1

Wishart model : Yj standard Gaussian in RM

sample covariance matrices Y = (Y1, . . . , YN) M × N matrix Y = (Yij)1≤i≤M,1≤j≤N Yij independent identically distributed (real or complex) E(Yij) = 0, E(Y 2

ij ) = 1

Wishart model : Yj standard Gaussian in RM numerous extensions

sample covariance matrices Y = (Y1, . . . , YN) M × N matrix Y = (Yij)1≤i≤M,1≤j≤N iid E(Yij) = 0, E(Y 2

ij ) = 1

sample covariance matrices Y = (Y1, . . . , YN) M × N matrix Y = (Yij)1≤i≤M,1≤j≤N iid E(Yij) = 0, E(Y 2

ij ) = 1

center of interest : eigenvalues 0 ≤ λN

1 ≤ · · · ≤ λN M

• f

Y Y t (M × M non-negative symmetric matrix)

sample covariance matrices Y = (Y1, . . . , YN) M × N matrix Y = (Yij)1≤i≤M,1≤j≤N iid E(Yij) = 0, E(Y 2

ij ) = 1

center of interest : eigenvalues 0 ≤ λN

1 ≤ · · · ≤ λN M

• f

Y Y t (M × M non-negative symmetric matrix)

• λN

k

singular values of Y

sample covariance matrices Y = (Y1, . . . , YN) M × N matrix Y = (Yij)1≤i≤M,1≤j≤N iid E(Yij) = 0, E(Y 2

ij ) = 1

center of interest : eigenvalues 0 ≤ λN

1 ≤ · · · ≤ λN M

• f

Y Y t (M × M non-negative symmetric matrix)

• λN

k

singular values of Y

• λN

k = λN k

N eigenvalues of 1 N Y Y t

sample covariance matrices Y = (Y1, . . . , YN) M × N matrix Y = (Yij)1≤i≤M,1≤j≤N iid E(Yij) = 0, E(Y 2

ij ) = 1

center of interest : eigenvalues 0 ≤ λN

1 ≤ · · · ≤ λN M

• f

Y Y t (M × M non-negative symmetric matrix)

• λN

k

singular values of Y

• λN

k = λN k

N eigenvalues of 1 N Y Y t spectral measure 1 M

M

• k=1

δ

λN

k

sample covariance matrices Y = (Y1, . . . , YN) M × N matrix Y = (Yij)1≤i≤M,1≤j≤N iid E(Yij) = 0, E(Y 2

ij ) = 1

center of interest : eigenvalues 0 ≤ λN

1 ≤ · · · ≤ λN M

• f

Y Y t (M × M non-negative symmetric matrix)

• λN

k

singular values of Y

• λN

k = λN k

N eigenvalues of 1 N Y Y t spectral measure 1 M

M

• k=1

δ

λN

k

asymptotics M = M(N) ∼ ρ N N → ∞

Marchenko-Pastur theorem (1967) asymptotic behavior of the spectral measure ( λ

N k = λN k /N)

Marchenko-Pastur theorem (1967) asymptotic behavior of the spectral measure ( λ

N k = λN k /N)

1 M

M

• k=1

δ

λN

k

→ ν Marchenko-Pastur distribution

Marchenko-Pastur theorem (1967) asymptotic behavior of the spectral measure ( λ

N k = λN k /N)

1 M

M

• k=1

δ

λN

k

→ ν Marchenko-Pastur distribution dν(x) =

• 1 − 1

ρ

• +δ0 +

1 ρ 2πx

• (b − x)(x − a) 1[a,b]dx

Marchenko-Pastur theorem (1967) asymptotic behavior of the spectral measure ( λ

N k = λN k /N)

1 M

M

• k=1

δ

λN

k

→ ν Marchenko-Pastur distribution dν(x) =

• 1 − 1

ρ

• +δ0 +

1 ρ 2πx

• (b − x)(x − a) 1[a,b]dx

a = a(ρ) =

• 1 − √ρ

2 b = b(ρ) =

• 1 + √ρ

2

Marchenko-Pastur theorem (1967) asymptotic behavior of the spectral measure ( λ

N k = λN k /N)

1 M

M

• k=1

δ

λN

k

→ ν Marchenko-Pastur distribution dν(x) =

• 1 − 1

ρ

• +δ0 +

1 ρ 2πx

• (b − x)(x − a) 1[a,b]dx

a = a(ρ) =

• 1 − √ρ

2 b = b(ρ) =

• 1 + √ρ

2

Marchenko-Pastur theorem 1 M

M

• k=1

δ

λN

k

→ ν

• n
• a(ρ), b(ρ)
• M ∼ ρ N

global regime

Marchenko-Pastur theorem 1 M

M

• k=1

δ

λN

k

→ ν

• n
• a(ρ), b(ρ)
• M ∼ ρ N

global regime large deviation asymptotics of the spectral measure

Marchenko-Pastur theorem 1 M

M

• k=1

δ

λN

k

→ ν

• n
• a(ρ), b(ρ)
• M ∼ ρ N

global regime large deviation asymptotics of the spectral measure fluctuations of the spectral measure

Marchenko-Pastur theorem 1 M

M

• k=1

δ

λN

k

→ ν

• n
• a(ρ), b(ρ)
• M ∼ ρ N

global regime large deviation asymptotics of the spectral measure fluctuations of the spectral measure

M

• k=1
• f
• λN

k

• −
• R f dν
• → G

Gaussian variable f : R → R smooth

Marchenko-Pastur theorem 1 M

M

• k=1

δ

λN

k

→ ν

• n
• a(ρ), b(ρ)
• M ∼ ρ N

local regime

Marchenko-Pastur theorem 1 M

M

• k=1

δ

λN

k

→ ν

• n
• a(ρ), b(ρ)
• M ∼ ρ N

local regime behavior of the individual eigenvalues

Marchenko-Pastur theorem 1 M

M

• k=1

δ

λN

k

→ ν

• n
• a(ρ), b(ρ)
• M ∼ ρ N

local regime behavior of the individual eigenvalues spacings (bulk behavior)

Marchenko-Pastur theorem 1 M

M

• k=1

δ

λN

k

→ ν

• n
• a(ρ), b(ρ)
• M ∼ ρ N

local regime behavior of the individual eigenvalues spacings (bulk behavior) extremal eigenvalues (edge behavior)

extremal eigenvalues largest eigenvalue λN

M = max1≤k≤M λN k

extremal eigenvalues largest eigenvalue λN

M = max1≤k≤M λN k

• λN

M = λN M

N

extremal eigenvalues largest eigenvalue λN

M = max1≤k≤M λN k

• λN

M = λN M

N → b(ρ) =

• 1 + √ρ

2 M ∼ ρ N

Marchenko-Pastur theorem (1967) asymptotic behavior of the spectral measure ( λ

N k = λN k /N)

1 M

M

• k=1

δ

λN

k

→ ν Marchenko-Pastur distribution dν(x) =

• 1 − 1

ρ

• +δ0 +

1 ρ 2πx

• (b − x)(x − a) 1[a,b]dx

a = a(ρ) =

• 1 − √ρ

2 b = b(ρ) =

• 1 + √ρ

2

extremal eigenvalues largest eigenvalue λN

M = max1≤k≤M λN k

• λN

M = λN M

N → b(ρ) =

• 1 + √ρ

2 M ∼ ρ N

extremal eigenvalues largest eigenvalue λN

M = max1≤k≤M λN k

• λN

M = λN M

N → b(ρ) =

• 1 + √ρ

2 M ∼ ρ N fluctuations around b(ρ)

extremal eigenvalues largest eigenvalue λN

M = max1≤k≤M λN k

• λN

M = λN M

N → b(ρ) =

• 1 + √ρ

2 M ∼ ρ N fluctuations around b(ρ) complex or real Gaussian (Wishart matrices)

extremal eigenvalues largest eigenvalue λN

M = max1≤k≤M λN k

• λN

M = λN M

N → b(ρ) =

• 1 + √ρ

2 M ∼ ρ N fluctuations around b(ρ) complex or real Gaussian (Wishart matrices) M2/3 λN

M − b(ρ)

• → C(ρ) FTW

extremal eigenvalues largest eigenvalue λN

M = max1≤k≤M λN k

• λN

M = λN M

N → b(ρ) =

• 1 + √ρ

2 M ∼ ρ N fluctuations around b(ρ) complex or real Gaussian (Wishart matrices) M2/3N−1 λN

M − b(ρ)N

• → C(ρ) FTW

extremal eigenvalues largest eigenvalue λN

M = max1≤k≤M λN k

• λN

M = λN M

N → b(ρ) =

• 1 + √ρ

2 M ∼ ρ N fluctuations around b(ρ) complex or real Gaussian (Wishart matrices) M2/3N−1 λN

M − b(ρ)N

• → C(ρ) FTW

FTW

• C. Tracy, H. Widom (1994) distribution

extremal eigenvalues largest eigenvalue λN

M = max1≤k≤M λN k

• λN

M = λN M

N → b(ρ) =

• 1 + √ρ

2 M ∼ ρ N fluctuations around b(ρ) complex or real Gaussian (Wishart matrices) M2/3N−1 λN

M − b(ρ)N

• → C(ρ) FTW

FTW

• C. Tracy, H. Widom (1994) distribution
• K. Johansson (2000), I. Johnstone (2001)

FTW

• C. Tracy, H. Widom (1994) distribution

(complex) FTW(s) = exp

• −

∞

s

(x − s)u(x)2dx

• ,

s ∈ R u′′ = 2u3 + xu Painlev´ e II equation

FTW

• C. Tracy, H. Widom (1994) distribution

(complex) FTW(s) = exp

• −

∞

s

(x − s)u(x)2dx

• ,

s ∈ R u′′ = 2u3 + xu Painlev´ e II equation density

mean ≃ −1.77 FTW(s) ∼ e−s3/12 as s → −∞ 1 − FTW(s) ∼ e−4s3/2/3 as s → +∞ density (similar for real case)

extremal eigenvalues largest eigenvalue λN

M = max1≤k≤M λN k

• λN

M = λN M

N → b(ρ) =

• 1 + √ρ

2 M ∼ ρ N fluctuations around b(ρ) complex or real Gaussian (Wishart matrices) M2/3 λN

M − b(ρ)

• → C(ρ) FTW

FTW

• C. Tracy, H. Widom (1994) distribution
• K. Johansson (2000), I. Johnstone (2001)

Gaussian (Wishart matrices)

Gaussian (Wishart matrices) completely solvable models

Gaussian (Wishart matrices) completely solvable models determinantal structure

• rthogonal polynomial analysis

Gaussian (Wishart matrices) completely solvable models determinantal structure

• rthogonal polynomial analysis

asymptotics of Laguerre orthogonal polynomials

Gaussian (Wishart matrices) completely solvable models determinantal structure

• rthogonal polynomial analysis

asymptotics of Laguerre orthogonal polynomials

• C. Tracy, H. Widom (1994)
• K. Johansson (2000), I. Johnstone (2001)

extension to non-Gaussian matrices

extension to non-Gaussian matrices

• A. Soshnikov (2001-02)

moment method E

• Tr
• (YY t)p

extension to non-Gaussian matrices

• A. Soshnikov (2001-02)

moment method E

• Tr
• (YY t)p
• L. Erd¨
• s, H.-T. Yau (2009-12) (and collaborators)

local Marchenko-Pastur law

• T. Tao, V. Vu (2010-11)

Lindeberg comparison method

extension to non-Gaussian matrices

• A. Soshnikov (2001-02)

moment method E

• Tr
• (YY t)p
• L. Erd¨
• s, H.-T. Yau (2009-12) (and collaborators)

local Marchenko-Pastur law

• T. Tao, V. Vu (2010-11)

Lindeberg comparison method symmetric matrices

(brief) survey of recent approaches to non-asymptotic exponential inequalities

(brief) survey of recent approaches to non-asymptotic exponential inequalities quantify the limit theorems

(brief) survey of recent approaches to non-asymptotic exponential inequalities quantify the limit theorems spectral measure

(brief) survey of recent approaches to non-asymptotic exponential inequalities quantify the limit theorems spectral measure extremal eigenvalues catch the new rate (mean)1/3

(brief) survey of recent approaches to non-asymptotic exponential inequalities quantify the limit theorems spectral measure extremal eigenvalues catch the new rate (mean)1/3 from the Gaussian case to non-Gaussian models

two main questions and objectives

two main questions and objectives tail inequalities for the spectral measure P M

• k=1

f ( λN

k ) ≥ t

Marchenko-Pastur theorem 1 M

M

• k=1

δ

λN

k

→ ν

• n
• a(ρ), b(ρ)
• M ∼ ρ N

global regime large deviation asymptotics of the spectral measure fluctuations of the spectral measure

M

• k=1
• f
• λN

k

• −
• R f dν
• → G

Gaussian variable f : R → R smooth

two main questions and objectives tail inequalities for the spectral measure P M

• k=1

f ( λN

k ) ≥ t

two main questions and objectives tail inequalities for the spectral measure P M

• k=1

f ( λN

k ) ≥ t

• tail inequalities for the extremal eigenvalues

P λN

M ≥ b(ρ) + ε

extremal eigenvalues largest eigenvalue λN

M = max1≤k≤M λN k

• λN

M = λN M

N → b(ρ) =

• 1 + √ρ

2 M ∼ ρ N fluctuations around b(ρ) complex or real Gaussian (Wishart matrices) M2/3 λN

M − b(ρ)

• → C(ρ) FTW

FTW

• C. Tracy, H. Widom (1994) distribution
• K. Johansson (2000), I. Johnstone (2001)

two main questions and objectives tail inequalities for the spectral measure P M

• k=1

f ( λN

k ) ≥ t

• tail inequalities for the extremal eigenvalues

P λN

M ≥ b(ρ) + ε

two main questions and objectives tail inequalities for the spectral measure P M

• k=1

f ( λN

k ) ≥ t

• tail inequalities for the extremal eigenvalues

P λN

M ≥ b(ρ) + ε

• Wishart matrices

more general covariance matrices

measure concentration tool

measure concentration tool F = F(Y Y t) = F(Yij)

measure concentration tool F = F(Y Y t) = F(Yij) satisfactory for the global regime

measure concentration tool F = F(Y Y t) = F(Yij) satisfactory for the global regime less satisfactory for the local regime

measure concentration tool F = F(Y Y t) = F(Yij) satisfactory for the global regime less satisfactory for the local regime specific functionals eigenvalue counting function extreme eigenvalues

two main questions and objectives tail inequalities for the spectral measure P M

• k=1

f ( λN

k ) ≥ t

• tail inequalities for the extremal eigenvalues

P λN

M ≥ b(ρ) + ε

• Wishart matrices

more general covariance matrices

tail inequalities for the spectral measure

• A. Guionnet, O. Zeitouni (2000)

measure concentration tool

tail inequalities for the spectral measure

• A. Guionnet, O. Zeitouni (2000)

measure concentration tool f : R → R smooth (Lipschitz)

tail inequalities for the spectral measure

• A. Guionnet, O. Zeitouni (2000)

measure concentration tool f : R → R smooth (Lipschitz) X = (Xij)1≤i,j≤M M × M symmetric matrix eigenvalues λ1 ≤ · · · ≤ λM

tail inequalities for the spectral measure

• A. Guionnet, O. Zeitouni (2000)

measure concentration tool f : R → R smooth (Lipschitz) X = (Xij)1≤i,j≤M M × M symmetric matrix eigenvalues λ1 ≤ · · · ≤ λM F : X → Tr f (X) =

M

• k=1

f (λk) Lipschitz with respect to the Euclidean structure on M × M matrices

tail inequalities for the spectral measure

• A. Guionnet, O. Zeitouni (2000)

measure concentration tool f : R → R smooth (Lipschitz) X = (Xij)1≤i,j≤M M × M symmetric matrix eigenvalues λ1 ≤ · · · ≤ λM F : X → Tr f (X) =

M

• k=1

f (λk) Lipschitz with respect to the Euclidean structure on M × M matrices convex if f is convex

concentration inequalities Sn =

1 √n (X1 + · · · + Xn)

F(X) = F(X1, . . . , Xn), F : Rn → R 1-Lipschitz X1, . . . , Xn independenty standard Gaussian P

• F(X) ≥ E
• F(X)
• + t
• ≤ e−t2/2,

t ≥ 0 0 ≤ Xi ≤ 1 independent, F 1-Lipschitz and convex P

• F(X) ≥ E
• F(X)
• + t
• ≤ 2 e−t2/4,

t ≥ 0

• M. Talagrand (1995)

tail inequalities for the spectral measure

tail inequalities for the spectral measure Gaussian entries Yij f : R → R such that f (x2) 1-Lipschitz

tail inequalities for the spectral measure Gaussian entries Yij f : R → R such that f (x2) 1-Lipschitz P M

• k=1
• f (

λN

k ) − E

• f (

λN

k )

• ≥ t
• ≤ C(ρ) e−t2/C(ρ),

t ≥ 0

tail inequalities for the spectral measure Gaussian entries Yij f : R → R such that f (x2) 1-Lipschitz P M

• k=1
• f (

λN

k ) − E

• f (

λN

k )

• ≥ t
• ≤ C(ρ) e−t2/C(ρ),

t ≥ 0 compactly supported entries Yij f : R → R such that f (x2) 1-Lipschitz and convex

Marchenko-Pastur theorem 1 M

M

• k=1

δ

λN

k

→ ν

• n
• a(ρ), b(ρ)
• M ∼ ρ N

global regime large deviation asymptotics of the spectral measure fluctuations of the spectral measure

M

• k=1
• f
• λN

k

• −
• R f dν
• → G

Gaussian variable f : R → R smooth

non-Lipschitz functions f

non-Lipschitz functions f typically f = 1I, I ⊂ R interval

M

• k=1

f

• λN

k

• = #
• λN

k ∈ I

• = NI

counting function

non-Lipschitz functions f typically f = 1I, I ⊂ R interval

M

• k=1

f

• λN

k

• = #
• λN

k ∈ I

• = NI

counting function Wishart matrices (determinantal structure)

non-Lipschitz functions f typically f = 1I, I ⊂ R interval

M

• k=1

f

• λN

k

• = #
• λN

k ∈ I

• = NI

counting function Wishart matrices (determinantal structure) I interval in (a, b) 1 √log M

• NI − E(NI)
• → G

Gaussian variable

non-Lipschitz functions f typically f = 1I, I ⊂ R interval

M

• k=1

f

• λN

k

• = #
• λN

k ∈ I

• = NI

counting function Wishart matrices (determinantal structure) I interval in (a, b) 1 √log M

• NI − E(NI)
• → G

Gaussian variable exponential tail inequalities P

• NI − E(NI) ≥ t
• ≤ C e−ct log(1+t/ log M),

t ≥ 0

non-Lipschitz functions f typically f = 1I, I ⊂ R interval

M

• k=1

f

• λN

k

• = #
• λN

k ∈ I

• = NI

counting function Wishart matrices (determinantal structure) I interval in (a, b) 1 √log M

• NI − E(NI)
• → G

Gaussian variable exponential tail inequalities P

• NI − E(NI) ≥ t
• ≤ C e−ct log(1+t/ log M),

t ≥ 0 Var

• NI
• = O(log M)

non-Gaussian covariance matrices comparison with Wishart model

non-Gaussian covariance matrices comparison with Wishart model partial results localization results

• L. Erd¨
• s, H.-T. Yau (2009-12)

Lindeberg comparison method

• T. Tao, V. Vu (2010-11)

non-Gaussian covariance matrices comparison with Wishart model partial results localization results

• L. Erd¨
• s, H.-T. Yau (2009-12)

Lindeberg comparison method

• T. Tao, V. Vu (2010-11)

Var

• NI
• = O(log M)
• S. Dallaporta, V. Vu (2011)

non-Gaussian covariance matrices comparison with Wishart model partial results localization results

• L. Erd¨
• s, H.-T. Yau (2009-12)

Lindeberg comparison method

• T. Tao, V. Vu (2010-11)

Var

• NI
• = O(log M)
• S. Dallaporta, V. Vu (2011)

P

• NI − E(NI) ≥ t
• ≤ C e−ctδ,

t ≥ C log M, 0 < δ ≤ 1

• T. Tao, V. Vu (2012)

non-Lipschitz functions f typically f = 1I, I ⊂ R interval

M

• k=1

f

• λN

k

• = #
• λN

k ∈ I

• = NI

counting function Wishart matrices (determinantal structure) I interval in (a, b) 1 √log M

• NI − E(NI)
• → G

Gaussian variable exponential tail inequalities P

• NI − E(NI) ≥ t
• ≤ C e−ct log(1+t/ log M),

t ≥ 0 Var

• NI
• = O(log M)

two main questions and objectives tail inequalities for the spectral measure P M

• k=1

f ( λN

k ) ≥ t

• tail inequalities for the extremal eigenvalues

P λN

M ≥ b(ρ) + ε

• Wishart matrices

more general covariance matrices

two main questions and objectives tail inequalities for the spectral measure P M

• k=1

f ( λN

k ) ≥ t

• tail inequalities for the extremal eigenvalues

P λN

M ≥ b(ρ) + ε

• Wishart matrices

more general covariance matrices

tail inequalities for the extremal eigenvalues fluctuations of the largest eigenvalue M2/3 λN

M − b(ρ)

• → C(ρ) FTW

M ∼ ρ N

extremal eigenvalues largest eigenvalue λN

M = max1≤k≤M λN k

• λN

M = λN M

N → b(ρ) =

• 1 + √ρ

2 M ∼ ρ N fluctuations around b(ρ) complex or real Gaussian (Wishart matrices) M2/3 λN

M − b(ρ)

• → C(ρ) FTW

FTW

• C. Tracy, H. Widom (1994) distribution
• K. Johansson (2000), I. Johnstone (2001)

tail inequalities for the extremal eigenvalues fluctuations of the largest eigenvalue M2/3 λN

M − b(ρ)

• → C(ρ) FTW

M ∼ ρ N

tail inequalities for the extremal eigenvalues fluctuations of the largest eigenvalue M2/3 λN

M − b(ρ)

• → C(ρ) FTW

M ∼ ρ N finite M inequalities

tail inequalities for the extremal eigenvalues fluctuations of the largest eigenvalue M2/3 λN

M − b(ρ)

• → C(ρ) FTW

M ∼ ρ N finite M inequalities at the (mean)1/3 rate reflecting the tails of FTW

tail inequalities for the extremal eigenvalues fluctuations of the largest eigenvalue M2/3 λN

M − b(ρ)

• → C(ρ) FTW

M ∼ ρ N finite M inequalities at the (mean)1/3 rate reflecting the tails of FTW bounds on Var( λN

M)

measure concentration tool

measure concentration tool (Gaussian) Wishart matrix Y Y t

measure concentration tool (Gaussian) Wishart matrix Y Y t λN

M = max 1≤k≤M λN k = sup |v|=1

|Y v|2 sN

M =

• λN

M

Lipschitz of the Gaussian entries Yij

measure concentration tool (Gaussian) Wishart matrix Y Y t λN

M = max 1≤k≤M λN k = sup |v|=1

|Y v|2 sN

M =

• λN

M

Lipschitz of the Gaussian entries Yij Gaussian concentration P

• sN

M ≥ E

• sN

M

• + t
• ≤ e−M t2/C,

t ≥ 0

measure concentration tool (Gaussian) Wishart matrix Y Y t λN

M = max 1≤k≤M λN k = sup |v|=1

|Y v|2 sN

M =

• λN

M

Lipschitz of the Gaussian entries Yij Gaussian concentration P

• sN

M ≥ E

• sN

M

• + t
• ≤ e−M t2/C,

t ≥ 0 E( sN

M) ∼

• b(ρ)

measure concentration tool (Gaussian) Wishart matrix Y Y t λN

M = max 1≤k≤M λN k = sup |v|=1

|Y v|2 sN

M =

• λN

M

Lipschitz of the Gaussian entries Yij Gaussian concentration P

• sN

M ≥ E

• sN

M

• + t
• ≤ e−M t2/C,

t ≥ 0 E( sN

M) ∼

• b(ρ)

correct large deviation bounds (t ≥ 1)

measure concentration tool (Gaussian) Wishart matrix Y Y t λN

M = max 1≤k≤M λN k = sup |v|=1

|Y v|2 sN

M =

• λN

M

Lipschitz of the Gaussian entries Yij Gaussian concentration P

• sN

M ≥ E

• sN

M

• + t
• ≤ e−M t2/C,

t ≥ 0 E( sN

M) ∼

• b(ρ)

does not fit the small deviation regime t = s M−2/3

extreme eigenvalues alternate tools

extreme eigenvalues alternate tools Riemann-Hilbert analysis (Wishart matrices) tri-diagonal representations (Wishart and β-ensembles) moment methods (Wishart and non-Gaussian matrices)

extreme eigenvalues alternate tools Riemann-Hilbert analysis (Wishart matrices) tri-diagonal representations (Wishart and β-ensembles) moment methods (Wishart and non-Gaussian matrices)

M2/3 λN

M − b(ρ)

• → C(ρ) FTW

P

• λN

M ≤ b(ρ) + s M−2/3

→ FTW(C s)

M2/3 λN

M − b(ρ)

• → C(ρ) FTW

P

• λN

M ≤ b(ρ) + s M−2/3

→ FTW(C s) bounds for Wishart matrices tri-diagonal representation

• B. Rider, M. L. (2010)

M2/3 λN

M − b(ρ)

• → C(ρ) FTW

P

• λN

M ≤ b(ρ) + s M−2/3

→ FTW(C s) bounds for Wishart matrices tri-diagonal representation

• B. Rider, M. L. (2010)

P λN

M ≥ b(ρ) + ǫ

• ≤ C e−Mε3/2/C,

0 < ε ≤ 1

M2/3 λN

M − b(ρ)

• → C(ρ) FTW

P

• λN

M ≤ b(ρ) + s M−2/3

→ FTW(C s) bounds for Wishart matrices tri-diagonal representation

• B. Rider, M. L. (2010)

P λN

M ≥ b(ρ) + ǫ

• ≤ C e−Mε3/2/C,

0 < ε ≤ 1 P λN

M ≤ b(ρ) − ǫ

• ≤ C e−Mε3/C,

0 < ε ≤ b(ρ)

P

• λN

M ≤ b(ρ) + s M−2/3

→ FTW(C s) bounds for Wishart matrices P λN

M ≥ b(ρ) + ǫ

• ≤ C e−Mε3/2/C,

0 < ε ≤ 1 P λN

M ≤ b(ρ) − ǫ

• ≤ C e−Mε3/C,

0 < ε ≤ b(ρ)

P

• λN

M ≤ b(ρ) + s M−2/3

→ FTW(C s) bounds for Wishart matrices P λN

M ≥ b(ρ) + ǫ

• ≤ C e−Mε3/2/C,

0 < ε ≤ 1 P λN

M ≤ b(ρ) − ǫ

• ≤ C e−Mε3/C,

0 < ε ≤ b(ρ) fit the Tracy-Widom asymptotics (ε = s M−2/3)

P

• λN

M ≤ b(ρ) + s M−2/3

→ FTW(C s) bounds for Wishart matrices P λN

M ≥ b(ρ) + ǫ

• ≤ C e−Mε3/2/C,

0 < ε ≤ 1 P λN

M ≤ b(ρ) − ǫ

• ≤ C e−Mε3/C,

0 < ε ≤ b(ρ) fit the Tracy-Widom asymptotics (ε = s M−2/3) 1 − FTW(s) ∼ e−s3/2/C (s → +∞) FTW(s) ∼ e−s3/C (s → −∞)

P

• λN

M ≤ b(ρ) + s M−2/3

→ FTW(C s) bounds for Wishart matrices P λN

M ≥ b(ρ) + ǫ

• ≤ C e−Mε3/2/C,

0 < ε ≤ 1 P λN

M ≤ b(ρ) − ǫ

• ≤ C e−Mε3/C,

0 < ε ≤ b(ρ) fit the Tracy-Widom asymptotics (ε = s M−2/3) 1 − FTW(s) ∼ e−s3/2/C (s → +∞) FTW(s) ∼ e−s3/C (s → −∞) Var( λN

M) = O

• 1

M4/3

M2/3 λN

M − b(ρ)

• → C(ρ) FTW

b(ρ) =

• 1 + √ρ

2

• λN

M = λN M/N,

M = M(N) ∼ ρ N ( √ MN)1/3 ( √ M + √ N)4/3

• λN

M − (

√ M + √ N)2 → FTW

M2/3 λN

M − b(ρ)

• → C(ρ) FTW

b(ρ) =

• 1 + √ρ

2

• λN

M = λN M/N,

M = M(N) ∼ ρ N ( √ MN)1/3 ( √ M + √ N)4/3

• λN

M − (

√ M + √ N)2 → FTW N + 1 ≥ M 0 < ε ≤ 1 P

• λN

M ≥ (

√ M + √ N)2(1 + ε)

• ≤ C e−

√ MN ε3/2( 1

√ε ∧

• M

N

1/4

)/C

P

• λN

M ≤ (

√ M + √ N)2(1 − ε)

• ≤ C e−MN ε3( 1

ε ∧

• M

N

1/2

)/C

bi and tri-diagonal representation

bi and tri-diagonal representation B =            χN · · · · · ·

• χ(M−1)

χN−1 · · · . . .

• χ(M−2)

χN−3 ... . . . . . . ... ... ... . . . · · · ...

• χ2

χN−M+2 · · · · · ·

• χ1

χN−M+1            χ(N−1), . . . , χ1,

• χ(M−1), . . . ,

χ1 independent chi-variables

bi and tri-diagonal representation B =            χN · · · · · ·

• χ(M−1)

χN−1 · · · . . .

• χ(M−2)

χN−3 ... . . . . . . ... ... ... . . . · · · ...

• χ2

χN−M+2 · · · · · ·

• χ1

χN−M+1            χ(N−1), . . . , χ1,

• χ(M−1), . . . ,

χ1 independent chi-variables B Bt same spectrum as Y Y t (Wishart)

bi and tri-diagonal representation B =            χN · · · · · ·

• χ(M−1)

χN−1 · · · . . .

• χ(M−2)

χN−3 ... . . . . . . ... ... ... . . . · · · ...

• χ2

χN−M+2 · · · · · ·

• χ1

χN−M+1            χ(N−1), . . . , χ1,

• χ(M−1), . . . ,

χ1 independent chi-variables B Bt same spectrum as Y Y t (Wishart)

• H. Trotter (1984), A. Edelman, I. Dimitriu (2002)

bi and tri-diagonal representation B =            χN · · · · · ·

• χ(M−1)

χN−1 · · · . . .

• χ(M−2)

χN−3 ... . . . . . . ... ... ... . . . · · · ...

• χ2

χN−M+2 · · · · · ·

• χ1

χN−M+1            χ(N−1), . . . , χ1,

• χ(M−1), . . . ,

χ1 independent chi-variables B Bt same spectrum as Y Y t (Wishart)

• H. Trotter (1984), A. Edelman, I. Dimitriu (2002)

extension to β-ensembles

bounds for non-Gaussian entries moment method E

• Tr
• (YY t)p
• O. Feldheim, S. Sodin (2010)

bounds for non-Gaussian entries moment method E

• Tr
• (YY t)p
• O. Feldheim, S. Sodin (2010)

largest eigenvalue (symmetric, subGaussian entries) P λN

M ≥ b(ρ) + ε

• ≤ C e−M ε3/2/C,

0 < ε ≤ 1

bounds for non-Gaussian entries moment method E

• Tr
• (YY t)p
• O. Feldheim, S. Sodin (2010)

largest eigenvalue (symmetric, subGaussian entries) P λN

M ≥ b(ρ) + ε

• ≤ C e−M ε3/2/C,

0 < ε ≤ 1 below the mean ?

bounds for non-Gaussian entries moment method E

• Tr
• (YY t)p
• O. Feldheim, S. Sodin (2010)

largest eigenvalue (symmetric, subGaussian entries) P λN

M ≥ b(ρ) + ε

• ≤ C e−M ε3/2/C,

0 < ε ≤ 1 below the mean ? necessary for variance bounds

variance level Var( λN

M) = O

• 1

M4/3

• S. Dallaporta (2012)

variance level Var( λN

M) = O

• 1

M4/3

• S. Dallaporta (2012)

comparison with Wishart model localization results L. Erd¨

• s, H.-T. Yau (2009-12)

Lindeberg comparison method T. Tao, V. Vu (2010-11)

smallest eigenvalue soft edge M = M(N) ∼ ρ N, ρ < 1 a(ρ) =

• 1 − √ρ

2

smallest eigenvalue soft edge M = M(N) ∼ ρ N, ρ < 1 a(ρ) =

• 1 − √ρ

2 P λN

1 ≤ a(ρ) − ε

• ≤ C e−M ε3/2/C,

0 < ε ≤ 1 P λN

1 ≥ a(ρ) + ε

• ≤ C e−M ε3/C,

0 < ε ≤ a(ρ) Wishart matrices

• B. Rider, M. L. (2010)

smallest eigenvalue hard edge M = N, ρ = 1 a(ρ) =

• 1 − √ρ

2 = 0

smallest eigenvalue hard edge M = N, ρ = 1 a(ρ) =

• 1 − √ρ

2 = 0 P

• λN

1 ≤ ε

N2

• ≤ C √ε + C e−cN

large families of covariance matrices

• M. Rudelson, R. Vershynin (2008-10)