concentration inequalities for random matrices · concentration inequalities for random matrices m....
TRANSCRIPT
![Page 1: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/1.jpg)
Concentration Inequalities for RandomMatrices
M. Ledoux
Institut de Mathematiques de Toulouse, France
![Page 2: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/2.jpg)
exponential tail inequalities
classical theme in probability and statistics
quantify the asymptotic statements
central limit theorems
large deviation principles
![Page 3: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/3.jpg)
exponential tail inequalities
classical theme in probability and statistics
quantify the asymptotic statements
central limit theorems
large deviation principles
![Page 4: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/4.jpg)
exponential tail inequalities
classical theme in probability and statistics
quantify the asymptotic statements
central limit theorems
large deviation principles
![Page 5: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/5.jpg)
classical exponential inequalities
sum of independent random variables
Sn =1√n
(X1 + · · ·+ Xn)
0 ≤ Xi ≤ 1 independent
P(Sn ≥ E(Sn) + t
)≤ e−t
2/2, t ≥ 0
Hoeffding’s inequality
same as for Xi standard Gaussian
central limit theorem
![Page 6: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/6.jpg)
classical exponential inequalities
sum of independent random variables
Sn =1√n
(X1 + · · ·+ Xn)
0 ≤ Xi ≤ 1 independent
P(Sn ≥ E(Sn) + t
)≤ e−t
2/2, t ≥ 0
Hoeffding’s inequality
same as for Xi standard Gaussian
central limit theorem
![Page 7: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/7.jpg)
classical exponential inequalities
sum of independent random variables
Sn =1√n
(X1 + · · ·+ Xn)
0 ≤ Xi ≤ 1 independent
P(Sn ≥ E(Sn) + t
)≤ e−t
2/2, t ≥ 0
Hoeffding’s inequality
same as for Xi standard Gaussian
central limit theorem
![Page 8: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/8.jpg)
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
![Page 9: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/9.jpg)
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
![Page 10: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/10.jpg)
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
![Page 11: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/11.jpg)
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
![Page 12: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/12.jpg)
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
![Page 13: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/13.jpg)
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−t
2/2, t ≥ 0
0 ≤ Xi ≤ 1 independent, F 1-Lipschitz and convex
P(F (X ) ≥ E
(F (X )
)+ t)≤ 2 e−t
2/4, t ≥ 0
M. Talagrand (1995)
![Page 14: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/14.jpg)
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−t
2/2, t ≥ 0
0 ≤ Xi ≤ 1 independent, F 1-Lipschitz and convex
P(F (X ) ≥ E
(F (X )
)+ t)≤ 2 e−t
2/4, t ≥ 0
M. Talagrand (1995)
![Page 15: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/15.jpg)
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−t
2/2, t ≥ 0
0 ≤ Xi ≤ 1 independent, F 1-Lipschitz
and convex
P(F (X ) ≥ E
(F (X )
)+ t)≤ 2 e−t
2/4, t ≥ 0
M. Talagrand (1995)
![Page 16: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/16.jpg)
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−t
2/2, t ≥ 0
0 ≤ Xi ≤ 1 independent, F 1-Lipschitz and convex
P(F (X ) ≥ E
(F (X )
)+ t)≤ 2 e−t
2/4, t ≥ 0
M. Talagrand (1995)
![Page 17: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/17.jpg)
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−t
2/2, t ≥ 0
0 ≤ Xi ≤ 1 independent, F 1-Lipschitz and convex
P(F (X ) ≥ E
(F (X )
)+ t)≤ 2 e−t
2/4, t ≥ 0
M. Talagrand (1995)
![Page 18: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/18.jpg)
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−t
2/2, t ≥ 0
0 ≤ Xi ≤ 1 independent, F 1-Lipschitz and convex
P(F (X ) ≥ E
(F (X )
)+ t)≤ 2 e−t
2/4, t ≥ 0
M. Talagrand (1995)
![Page 19: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/19.jpg)
empirical processes
X1, . . . ,Xn independent with values in (S ,S)
F collection of functions f : S → [0, 1]
Z = supf ∈F
n∑i=1
f (Xi )
Z Lipschitz and convex
concentration inequalities on
P(∣∣Z − E(Z )
∣∣ ≥ t), t ≥ 0
![Page 20: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/20.jpg)
empirical processes
X1, . . . ,Xn independent with values in (S ,S)
F collection of functions f : S → [0, 1]
Z = supf ∈F
n∑i=1
f (Xi )
Z Lipschitz and convex
concentration inequalities on
P(∣∣Z − E(Z )
∣∣ ≥ t), t ≥ 0
![Page 21: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/21.jpg)
empirical processes
X1, . . . ,Xn independent with values in (S ,S)
F collection of functions f : S → [0, 1]
Z = supf ∈F
n∑i=1
f (Xi )
Z Lipschitz and convex
concentration inequalities on
P(∣∣Z − E(Z )
∣∣ ≥ t), t ≥ 0
![Page 22: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/22.jpg)
empirical processes
X1, . . . ,Xn independent with values in (S ,S)
F collection of functions f : S → [0, 1]
Z = supf ∈F
n∑i=1
f (Xi )
Z Lipschitz and convex
concentration inequalities on
P(∣∣Z − E(Z )
∣∣ ≥ t), t ≥ 0
![Page 23: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/23.jpg)
Z = supf ∈F
n∑i=1
f (Xi )
|f | ≤ 1, E(f (Xi )) = 0, f ∈ F
P(|Z −M| ≥ t
)≤ C exp
(− t
Clog
(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)
![Page 24: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/24.jpg)
Z = supf ∈F
n∑i=1
f (Xi )
|f | ≤ 1, E(f (Xi )) = 0, f ∈ F
P(|Z −M| ≥ t
)≤ C exp
(− t
Clog
(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)
![Page 25: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/25.jpg)
Z = supf ∈F
n∑i=1
f (Xi )
|f | ≤ 1, E(f (Xi )) = 0, f ∈ F
P(|Z −M| ≥ t
)≤ C exp
(− t
Clog
(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)
![Page 26: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/26.jpg)
Z = supf ∈F
n∑i=1
f (Xi )
|f | ≤ 1, E(f (Xi )) = 0, f ∈ F
P(|Z −M| ≥ t
)≤ C exp
(− t
Clog
(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)
![Page 27: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/27.jpg)
Z = supf ∈F
n∑i=1
f (Xi )
|f | ≤ 1, E(f (Xi )) = 0, f ∈ F
P(|Z −M| ≥ t
)≤ C exp
(− t
Clog
(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)
![Page 28: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/28.jpg)
concentration inequalities
numerous applications
• geometric functional analysis
• discrete and combinatorial probability
• empirical processes
• statistical mechanics
• random matrix theory
![Page 29: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/29.jpg)
concentration inequalities
numerous applications
• geometric functional analysis
• discrete and combinatorial probability
• empirical processes
• statistical mechanics
• random matrix theory
![Page 30: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/30.jpg)
concentration inequalities
numerous applications
• geometric functional analysis
• discrete and combinatorial probability
• empirical processes
• statistical mechanics
• random matrix theory
![Page 31: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/31.jpg)
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...
![Page 32: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/32.jpg)
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...
![Page 33: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/33.jpg)
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...
![Page 34: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/34.jpg)
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...
![Page 35: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/35.jpg)
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...
![Page 36: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/36.jpg)
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
![Page 37: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/37.jpg)
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
![Page 38: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/38.jpg)
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
![Page 39: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/39.jpg)
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
![Page 40: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/40.jpg)
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
![Page 41: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/41.jpg)
is Y Y t a good approximation of the
population covariance matrix
E(Y Y t) ?
M finite
1
NY Y t → E(Y Y t) N →∞
M infinite ?
M = M(N) → ∞ N →∞
M
N∼ ρ ∈ (0,∞) N →∞
![Page 42: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/42.jpg)
is Y Y t a good approximation of the
population covariance matrix
E(Y Y t) ?
M finite
1
NY Y t → E(Y Y t) N →∞
M infinite ?
M = M(N) → ∞ N →∞
M
N∼ ρ ∈ (0,∞) N →∞
![Page 43: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/43.jpg)
is Y Y t a good approximation of the
population covariance matrix
E(Y Y t) ?
M finite
1
NY Y t → E(Y Y t) N →∞
M infinite ?
M = M(N) → ∞ N →∞
M
N∼ ρ ∈ (0,∞) N →∞
![Page 44: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/44.jpg)
is Y Y t a good approximation of the
population covariance matrix
E(Y Y t) ?
M finite
1
NY Y t → E(Y Y t) N →∞
M infinite ?
M = M(N) → ∞ N →∞
M
N∼ ρ ∈ (0,∞) N →∞
![Page 45: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/45.jpg)
is Y Y t a good approximation of the
population covariance matrix
E(Y Y t) ?
M finite
1
NY Y t → E(Y Y t) N →∞
M infinite ?
M = M(N) → ∞ N →∞
M
N∼ ρ ∈ (0,∞) N →∞
![Page 46: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/46.jpg)
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 2ij ) = 1
Wishart model : Yj standard Gaussian in RM
numerous extensions
![Page 47: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/47.jpg)
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 2ij ) = 1
Wishart model : Yj standard Gaussian in RM
numerous extensions
![Page 48: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/48.jpg)
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 2ij ) = 1
Wishart model : Yj standard Gaussian in RM
numerous extensions
![Page 49: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/49.jpg)
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 2ij ) = 1
Wishart model : Yj standard Gaussian in RM
numerous extensions
![Page 50: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/50.jpg)
sample covariance matrices
Y = (Y1, . . . ,YN) M × N matrix
Y = (Yij)1≤i≤M,1≤j≤N iid E(Yij) = 0, E(Y 2ij ) = 1
center of interest : eigenvalues 0 ≤ λN1 ≤ · · · ≤ λNM
of Y Y t (M ×M non-negative symmetric matrix)√λNk singular values of Y
λNk =λNkN
eigenvalues of1
NY Y t
spectral measure1
M
M∑k=1
δλNk
asymptotics M = M(N) ∼ ρN N →∞
![Page 51: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/51.jpg)
sample covariance matrices
Y = (Y1, . . . ,YN) M × N matrix
Y = (Yij)1≤i≤M,1≤j≤N iid E(Yij) = 0, E(Y 2ij ) = 1
center of interest : eigenvalues 0 ≤ λN1 ≤ · · · ≤ λNM
of Y Y t (M ×M non-negative symmetric matrix)
√λNk singular values of Y
λNk =λNkN
eigenvalues of1
NY Y t
spectral measure1
M
M∑k=1
δλNk
asymptotics M = M(N) ∼ ρN N →∞
![Page 52: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/52.jpg)
sample covariance matrices
Y = (Y1, . . . ,YN) M × N matrix
Y = (Yij)1≤i≤M,1≤j≤N iid E(Yij) = 0, E(Y 2ij ) = 1
center of interest : eigenvalues 0 ≤ λN1 ≤ · · · ≤ λNM
of Y Y t (M ×M non-negative symmetric matrix)√λNk singular values of Y
λNk =λNkN
eigenvalues of1
NY Y t
spectral measure1
M
M∑k=1
δλNk
asymptotics M = M(N) ∼ ρN N →∞
![Page 53: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/53.jpg)
sample covariance matrices
Y = (Y1, . . . ,YN) M × N matrix
Y = (Yij)1≤i≤M,1≤j≤N iid E(Yij) = 0, E(Y 2ij ) = 1
center of interest : eigenvalues 0 ≤ λN1 ≤ · · · ≤ λNM
of Y Y t (M ×M non-negative symmetric matrix)√λNk singular values of Y
λNk =λNkN
eigenvalues of1
NY Y t
spectral measure1
M
M∑k=1
δλNk
asymptotics M = M(N) ∼ ρN N →∞
![Page 54: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/54.jpg)
sample covariance matrices
Y = (Y1, . . . ,YN) M × N matrix
Y = (Yij)1≤i≤M,1≤j≤N iid E(Yij) = 0, E(Y 2ij ) = 1
center of interest : eigenvalues 0 ≤ λN1 ≤ · · · ≤ λNM
of Y Y t (M ×M non-negative symmetric matrix)√λNk singular values of Y
λNk =λNkN
eigenvalues of1
NY Y t
spectral measure1
M
M∑k=1
δλNk
asymptotics M = M(N) ∼ ρN N →∞
![Page 55: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/55.jpg)
sample covariance matrices
Y = (Y1, . . . ,YN) M × N matrix
Y = (Yij)1≤i≤M,1≤j≤N iid E(Yij) = 0, E(Y 2ij ) = 1
center of interest : eigenvalues 0 ≤ λN1 ≤ · · · ≤ λNM
of Y Y t (M ×M non-negative symmetric matrix)√λNk singular values of Y
λNk =λNkN
eigenvalues of1
NY Y t
spectral measure1
M
M∑k=1
δλNk
asymptotics M = M(N) ∼ ρN N →∞
![Page 56: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/56.jpg)
Marchenko-Pastur theorem (1967)
asymptotic behavior of the spectral measure (λN
k = λNk /N)
1
M
M∑k=1
δλNk→ 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
)2b = b(ρ) =
(1 +√ρ)2
![Page 57: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/57.jpg)
Marchenko-Pastur theorem (1967)
asymptotic behavior of the spectral measure (λN
k = λNk /N)
1
M
M∑k=1
δλNk→ 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
)2b = b(ρ) =
(1 +√ρ)2
![Page 58: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/58.jpg)
Marchenko-Pastur theorem (1967)
asymptotic behavior of the spectral measure (λN
k = λNk /N)
1
M
M∑k=1
δλNk→ 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
)2b = b(ρ) =
(1 +√ρ)2
![Page 59: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/59.jpg)
Marchenko-Pastur theorem (1967)
asymptotic behavior of the spectral measure (λN
k = λNk /N)
1
M
M∑k=1
δλNk→ 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
)2b = b(ρ) =
(1 +√ρ)2
![Page 60: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/60.jpg)
Marchenko-Pastur theorem (1967)
asymptotic behavior of the spectral measure (λN
k = λNk /N)
1
M
M∑k=1
δλNk→ 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
)2b = b(ρ) =
(1 +√ρ)2
![Page 61: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/61.jpg)
Marchenko-Pastur theorem
1
M
M∑k=1
δλNk→ ν on
(a(ρ), b(ρ)
)M ∼ ρN
global regime
large deviation asymptotics of the spectral measure
fluctuations of the spectral measure
M∑k=1
[f(λNk)−∫R f dν
]→ G Gaussian variable
f : R→ R smooth
![Page 62: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/62.jpg)
Marchenko-Pastur theorem
1
M
M∑k=1
δλNk→ ν on
(a(ρ), b(ρ)
)M ∼ ρN
global regime
large deviation asymptotics of the spectral measure
fluctuations of the spectral measure
M∑k=1
[f(λNk)−∫R f dν
]→ G Gaussian variable
f : R→ R smooth
![Page 63: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/63.jpg)
Marchenko-Pastur theorem
1
M
M∑k=1
δλNk→ ν on
(a(ρ), b(ρ)
)M ∼ ρN
global regime
large deviation asymptotics of the spectral measure
fluctuations of the spectral measure
M∑k=1
[f(λNk)−∫R f dν
]→ G Gaussian variable
f : R→ R smooth
![Page 64: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/64.jpg)
Marchenko-Pastur theorem
1
M
M∑k=1
δλNk→ ν on
(a(ρ), b(ρ)
)M ∼ ρN
global regime
large deviation asymptotics of the spectral measure
fluctuations of the spectral measure
M∑k=1
[f(λNk)−∫R f dν
]→ G Gaussian variable
f : R→ R smooth
![Page 65: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/65.jpg)
Marchenko-Pastur theorem
1
M
M∑k=1
δλNk→ ν on
(a(ρ), b(ρ)
)M ∼ ρN
local regime
behavior of the individual eigenvalues
spacings (bulk behavior)
extremal eigenvalues (edge behavior)
![Page 66: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/66.jpg)
Marchenko-Pastur theorem
1
M
M∑k=1
δλNk→ ν on
(a(ρ), b(ρ)
)M ∼ ρN
local regime
behavior of the individual eigenvalues
spacings (bulk behavior)
extremal eigenvalues (edge behavior)
![Page 67: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/67.jpg)
Marchenko-Pastur theorem
1
M
M∑k=1
δλNk→ ν on
(a(ρ), b(ρ)
)M ∼ ρN
local regime
behavior of the individual eigenvalues
spacings (bulk behavior)
extremal eigenvalues (edge behavior)
![Page 68: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/68.jpg)
Marchenko-Pastur theorem
1
M
M∑k=1
δλNk→ ν on
(a(ρ), b(ρ)
)M ∼ ρN
local regime
behavior of the individual eigenvalues
spacings (bulk behavior)
extremal eigenvalues (edge behavior)
![Page 69: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/69.jpg)
extremal eigenvalues
largest eigenvalue λNM = max1≤k≤M λNk
λNM =λNMN→ b(ρ) =
(1 +√ρ)2
M ∼ ρN
![Page 70: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/70.jpg)
extremal eigenvalues
largest eigenvalue λNM = max1≤k≤M λNk
λNM =λNMN
→ b(ρ) =(1 +√ρ)2
M ∼ ρN
![Page 71: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/71.jpg)
extremal eigenvalues
largest eigenvalue λNM = max1≤k≤M λNk
λNM =λNMN→ b(ρ) =
(1 +√ρ)2
M ∼ ρN
![Page 72: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/72.jpg)
Marchenko-Pastur theorem (1967)
asymptotic behavior of the spectral measure (λN
k = λNk /N)
1
M
M∑k=1
δλNk→ 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
)2b = b(ρ) =
(1 +√ρ)2
![Page 73: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/73.jpg)
extremal eigenvalues
largest eigenvalue λNM = max1≤k≤M λNk
λNM =λNMN→ b(ρ) =
(1 +√ρ)2
M ∼ ρN
fluctuations around b(ρ)
complex or real Gaussian (Wishart matrices)
FTW C. Tracy, H. Widom (1994) distribution
K. Johansson (2000), I. Johnstone (2001)
![Page 74: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/74.jpg)
extremal eigenvalues
largest eigenvalue λNM = max1≤k≤M λNk
λNM =λNMN→ b(ρ) =
(1 +√ρ)2
M ∼ ρN
fluctuations around b(ρ)
complex or real Gaussian (Wishart matrices)
FTW C. Tracy, H. Widom (1994) distribution
K. Johansson (2000), I. Johnstone (2001)
![Page 75: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/75.jpg)
extremal eigenvalues
largest eigenvalue λNM = max1≤k≤M λNk
λNM =λNMN→ b(ρ) =
(1 +√ρ)2
M ∼ ρN
fluctuations around b(ρ)
complex or real Gaussian (Wishart matrices)
FTW C. Tracy, H. Widom (1994) distribution
K. Johansson (2000), I. Johnstone (2001)
![Page 76: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/76.jpg)
extremal eigenvalues
largest eigenvalue λNM = max1≤k≤M λNk
λNM =λNMN→ b(ρ) =
(1 +√ρ)2
M ∼ ρN
fluctuations around b(ρ)
complex or real Gaussian (Wishart matrices)
M2/3[λNM − b(ρ)
]→ C (ρ)FTW
FTW C. Tracy, H. Widom (1994) distribution
K. Johansson (2000), I. Johnstone (2001)
![Page 77: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/77.jpg)
extremal eigenvalues
largest eigenvalue λNM = max1≤k≤M λNk
λNM =λNMN→ b(ρ) =
(1 +√ρ)2
M ∼ ρN
fluctuations around b(ρ)
complex or real Gaussian (Wishart matrices)
M2/3N−1[λNM − b(ρ)N
]→ C (ρ)FTW
FTW C. Tracy, H. Widom (1994) distribution
K. Johansson (2000), I. Johnstone (2001)
![Page 78: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/78.jpg)
extremal eigenvalues
largest eigenvalue λNM = max1≤k≤M λNk
λNM =λNMN→ b(ρ) =
(1 +√ρ)2
M ∼ ρN
fluctuations around b(ρ)
complex or real Gaussian (Wishart matrices)
M2/3N−1[λNM − b(ρ)N
]→ C (ρ)FTW
FTW C. Tracy, H. Widom (1994) distribution
K. Johansson (2000), I. Johnstone (2001)
![Page 79: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/79.jpg)
extremal eigenvalues
largest eigenvalue λNM = max1≤k≤M λNk
λNM =λNMN→ b(ρ) =
(1 +√ρ)2
M ∼ ρN
fluctuations around b(ρ)
complex or real Gaussian (Wishart matrices)
M2/3N−1[λNM − b(ρ)N
]→ C (ρ)FTW
FTW C. Tracy, H. Widom (1994) distribution
K. Johansson (2000), I. Johnstone (2001)
![Page 80: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/80.jpg)
FTW C. Tracy, H. Widom (1994) distribution
(complex) FTW(s) = exp
(−∫ ∞s
(x − s)u(x)2dx
), s ∈ R
u′′ = 2u3 + xu Painleve II equation
density
![Page 81: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/81.jpg)
FTW C. Tracy, H. Widom (1994) distribution
(complex) FTW(s) = exp
(−∫ ∞s
(x − s)u(x)2dx
), s ∈ R
u′′ = 2u3 + xu Painleve II equation
density
![Page 82: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/82.jpg)
mean ' −1.77
FTW(s) ∼ e−s3/12 as s → −∞
1− FTW(s) ∼ e−4s3/2/3 as s → +∞
density (similar for real case)
![Page 83: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/83.jpg)
extremal eigenvalues
largest eigenvalue λNM = max1≤k≤M λNk
λNM =λNMN→ b(ρ) =
(1 +√ρ)2
M ∼ ρN
fluctuations around b(ρ)
complex or real Gaussian (Wishart matrices)
M2/3[λNM − b(ρ)
]→ C (ρ)FTW
FTW C. Tracy, H. Widom (1994) distribution
K. Johansson (2000), I. Johnstone (2001)
![Page 84: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/84.jpg)
Gaussian (Wishart matrices)
completely solvable models
determinantal structure
orthogonal polynomial analysis
asymptotics of Laguerre orthogonal polynomials
C. Tracy, H. Widom (1994)
K. Johansson (2000), I. Johnstone (2001)
![Page 85: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/85.jpg)
Gaussian (Wishart matrices)
completely solvable models
determinantal structure
orthogonal polynomial analysis
asymptotics of Laguerre orthogonal polynomials
C. Tracy, H. Widom (1994)
K. Johansson (2000), I. Johnstone (2001)
![Page 86: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/86.jpg)
Gaussian (Wishart matrices)
completely solvable models
determinantal structure
orthogonal polynomial analysis
asymptotics of Laguerre orthogonal polynomials
C. Tracy, H. Widom (1994)
K. Johansson (2000), I. Johnstone (2001)
![Page 87: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/87.jpg)
Gaussian (Wishart matrices)
completely solvable models
determinantal structure
orthogonal polynomial analysis
asymptotics of Laguerre orthogonal polynomials
C. Tracy, H. Widom (1994)
K. Johansson (2000), I. Johnstone (2001)
![Page 88: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/88.jpg)
Gaussian (Wishart matrices)
completely solvable models
determinantal structure
orthogonal polynomial analysis
asymptotics of Laguerre orthogonal polynomials
C. Tracy, H. Widom (1994)
K. Johansson (2000), I. Johnstone (2001)
![Page 89: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/89.jpg)
extension to non-Gaussian matrices
A. Soshnikov (2001-02)
moment method E(Tr((YY t)p
))L. Erdos, H.-T. Yau (2009-12) (and collaborators)
local Marchenko-Pastur law
T. Tao, V. Vu (2010-11)
Lindeberg comparison method
symmetric matrices
![Page 90: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/90.jpg)
extension to non-Gaussian matrices
A. Soshnikov (2001-02)
moment method E(Tr((YY t)p
))
L. Erdos, H.-T. Yau (2009-12) (and collaborators)
local Marchenko-Pastur law
T. Tao, V. Vu (2010-11)
Lindeberg comparison method
symmetric matrices
![Page 91: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/91.jpg)
extension to non-Gaussian matrices
A. Soshnikov (2001-02)
moment method E(Tr((YY t)p
))L. Erdos, H.-T. Yau (2009-12) (and collaborators)
local Marchenko-Pastur law
T. Tao, V. Vu (2010-11)
Lindeberg comparison method
symmetric matrices
![Page 92: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/92.jpg)
extension to non-Gaussian matrices
A. Soshnikov (2001-02)
moment method E(Tr((YY t)p
))L. Erdos, H.-T. Yau (2009-12) (and collaborators)
local Marchenko-Pastur law
T. Tao, V. Vu (2010-11)
Lindeberg comparison method
symmetric matrices
![Page 93: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/93.jpg)
(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
![Page 94: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/94.jpg)
(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
![Page 95: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/95.jpg)
(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
![Page 96: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/96.jpg)
(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
![Page 97: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/97.jpg)
(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
![Page 98: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/98.jpg)
two main questions and objectives
tail inequalities for the spectral measure
P( M∑
k=1
f (λNk ) ≥ t
)
![Page 99: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/99.jpg)
two main questions and objectives
tail inequalities for the spectral measure
P( M∑
k=1
f (λNk ) ≥ t
)
![Page 100: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/100.jpg)
Marchenko-Pastur theorem
1
M
M∑k=1
δλNk→ ν on
(a(ρ), b(ρ)
)M ∼ ρN
global regime
large deviation asymptotics of the spectral measure
fluctuations of the spectral measure
M∑k=1
[f(λNk)−∫R f dν
]→ G Gaussian variable
f : R→ R smooth
![Page 101: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/101.jpg)
two main questions and objectives
tail inequalities for the spectral measure
P( M∑
k=1
f (λNk ) ≥ t
)
tail inequalities for the extremal eigenvalues
P(λNM ≥ b(ρ) + ε
)
![Page 102: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/102.jpg)
two main questions and objectives
tail inequalities for the spectral measure
P( M∑
k=1
f (λNk ) ≥ t
)
tail inequalities for the extremal eigenvalues
P(λNM ≥ b(ρ) + ε
)
![Page 103: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/103.jpg)
extremal eigenvalues
largest eigenvalue λNM = max1≤k≤M λNk
λNM =λNMN→ b(ρ) =
(1 +√ρ)2
M ∼ ρN
fluctuations around b(ρ)
complex or real Gaussian (Wishart matrices)
M2/3[λNM − b(ρ)
]→ C (ρ)FTW
FTW C. Tracy, H. Widom (1994) distribution
K. Johansson (2000), I. Johnstone (2001)
![Page 104: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/104.jpg)
two main questions and objectives
tail inequalities for the spectral measure
P( M∑
k=1
f (λNk ) ≥ t
)
tail inequalities for the extremal eigenvalues
P(λNM ≥ b(ρ) + ε
)
Wishart matrices
more general covariance matrices
![Page 105: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/105.jpg)
two main questions and objectives
tail inequalities for the spectral measure
P( M∑
k=1
f (λNk ) ≥ t
)
tail inequalities for the extremal eigenvalues
P(λNM ≥ b(ρ) + ε
)
Wishart matrices
more general covariance matrices
![Page 106: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/106.jpg)
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
![Page 107: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/107.jpg)
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
![Page 108: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/108.jpg)
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
![Page 109: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/109.jpg)
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
![Page 110: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/110.jpg)
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
![Page 111: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/111.jpg)
two main questions and objectives
tail inequalities for the spectral measure
P( M∑
k=1
f (λNk ) ≥ t
)
tail inequalities for the extremal eigenvalues
P(λNM ≥ b(ρ) + ε
)
Wishart matrices
more general covariance matrices
![Page 112: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/112.jpg)
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
![Page 113: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/113.jpg)
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
![Page 114: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/114.jpg)
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
![Page 115: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/115.jpg)
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
![Page 116: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/116.jpg)
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
![Page 117: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/117.jpg)
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−t
2/2, t ≥ 0
0 ≤ Xi ≤ 1 independent, F 1-Lipschitz and convex
P(F (X ) ≥ E
(F (X )
)+ t)≤ 2 e−t
2/4, t ≥ 0
M. Talagrand (1995)
![Page 118: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/118.jpg)
tail inequalities for the spectral measure
Gaussian entries Yij
f : R→ R such that f (x2) 1-Lipschitz
P( M∑
k=1
[f (λNk )− E
(f (λNk )
)]≥ t
)≤ C (ρ) e−t
2/C(ρ), t ≥ 0
compactly supported entries Yij
f : R→ R such that f (x2) 1-Lipschitz and convex
![Page 119: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/119.jpg)
tail inequalities for the spectral measure
Gaussian entries Yij
f : R→ R such that f (x2) 1-Lipschitz
P( M∑
k=1
[f (λNk )− E
(f (λNk )
)]≥ t
)≤ C (ρ) e−t
2/C(ρ), t ≥ 0
compactly supported entries Yij
f : R→ R such that f (x2) 1-Lipschitz and convex
![Page 120: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/120.jpg)
tail inequalities for the spectral measure
Gaussian entries Yij
f : R→ R such that f (x2) 1-Lipschitz
P( M∑
k=1
[f (λNk )− E
(f (λNk )
)]≥ t
)≤ C (ρ) e−t
2/C(ρ), t ≥ 0
compactly supported entries Yij
f : R→ R such that f (x2) 1-Lipschitz and convex
![Page 121: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/121.jpg)
tail inequalities for the spectral measure
Gaussian entries Yij
f : R→ R such that f (x2) 1-Lipschitz
P( M∑
k=1
[f (λNk )− E
(f (λNk )
)]≥ t
)≤ C (ρ) e−t
2/C(ρ), t ≥ 0
compactly supported entries Yij
f : R→ R such that f (x2) 1-Lipschitz and convex
![Page 122: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/122.jpg)
Marchenko-Pastur theorem
1
M
M∑k=1
δλNk→ ν on
(a(ρ), b(ρ)
)M ∼ ρN
global regime
large deviation asymptotics of the spectral measure
fluctuations of the spectral measure
M∑k=1
[f(λNk)−∫R f dν
]→ G Gaussian variable
f : R→ R smooth
![Page 123: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/123.jpg)
non-Lipschitz functions f
typically f = 1I , I ⊂ R interval
M∑k=1
f(λNk)
= #{λNk ∈ I
}= NI counting function
Wishart matrices (determinantal structure)
I interval in (a, b)
1√logM
[NI − E(NI )
]→ G Gaussian variable
exponential tail inequalities
P(NI − E(NI ) ≥ t
)≤ C e−ct log(1+t/ logM), t ≥ 0
Var(NI
)= O(logM)
![Page 124: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/124.jpg)
non-Lipschitz functions f
typically f = 1I , I ⊂ R interval
M∑k=1
f(λNk)
= #{λNk ∈ I
}= NI counting function
Wishart matrices (determinantal structure)
I interval in (a, b)
1√logM
[NI − E(NI )
]→ G Gaussian variable
exponential tail inequalities
P(NI − E(NI ) ≥ t
)≤ C e−ct log(1+t/ logM), t ≥ 0
Var(NI
)= O(logM)
![Page 125: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/125.jpg)
non-Lipschitz functions f
typically f = 1I , I ⊂ R interval
M∑k=1
f(λNk)
= #{λNk ∈ I
}= NI counting function
Wishart matrices (determinantal structure)
I interval in (a, b)
1√logM
[NI − E(NI )
]→ G Gaussian variable
exponential tail inequalities
P(NI − E(NI ) ≥ t
)≤ C e−ct log(1+t/ logM), t ≥ 0
Var(NI
)= O(logM)
![Page 126: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/126.jpg)
non-Lipschitz functions f
typically f = 1I , I ⊂ R interval
M∑k=1
f(λNk)
= #{λNk ∈ I
}= NI counting function
Wishart matrices (determinantal structure)
I interval in (a, b)
1√logM
[NI − E(NI )
]→ G Gaussian variable
exponential tail inequalities
P(NI − E(NI ) ≥ t
)≤ C e−ct log(1+t/ logM), t ≥ 0
Var(NI
)= O(logM)
![Page 127: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/127.jpg)
non-Lipschitz functions f
typically f = 1I , I ⊂ R interval
M∑k=1
f(λNk)
= #{λNk ∈ I
}= NI counting function
Wishart matrices (determinantal structure)
I interval in (a, b)
1√logM
[NI − E(NI )
]→ G Gaussian variable
exponential tail inequalities
P(NI − E(NI ) ≥ t
)≤ C e−ct log(1+t/ logM), t ≥ 0
Var(NI
)= O(logM)
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non-Lipschitz functions f
typically f = 1I , I ⊂ R interval
M∑k=1
f(λNk)
= #{λNk ∈ I
}= NI counting function
Wishart matrices (determinantal structure)
I interval in (a, b)
1√logM
[NI − E(NI )
]→ G Gaussian variable
exponential tail inequalities
P(NI − E(NI ) ≥ t
)≤ C e−ct log(1+t/ logM), t ≥ 0
Var(NI
)= O(logM)
![Page 129: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/129.jpg)
non-Gaussian covariance matrices
comparison with Wishart model
partial results
localization results L. Erdos, H.-T. Yau (2009-12)
Lindeberg comparison method T. Tao, V. Vu (2010-11)
Var(NI
)= O(logM)
S. Dallaporta, V. Vu (2011)
P(NI − E(NI ) ≥ t
)≤ C e−ct
δ, t ≥ C logM, 0 < δ ≤ 1
T. Tao, V. Vu (2012)
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non-Gaussian covariance matrices
comparison with Wishart model
partial results
localization results L. Erdos, H.-T. Yau (2009-12)
Lindeberg comparison method T. Tao, V. Vu (2010-11)
Var(NI
)= O(logM)
S. Dallaporta, V. Vu (2011)
P(NI − E(NI ) ≥ t
)≤ C e−ct
δ, t ≥ C logM, 0 < δ ≤ 1
T. Tao, V. Vu (2012)
![Page 131: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/131.jpg)
non-Gaussian covariance matrices
comparison with Wishart model
partial results
localization results L. Erdos, H.-T. Yau (2009-12)
Lindeberg comparison method T. Tao, V. Vu (2010-11)
Var(NI
)= O(logM)
S. Dallaporta, V. Vu (2011)
P(NI − E(NI ) ≥ t
)≤ C e−ct
δ, t ≥ C logM, 0 < δ ≤ 1
T. Tao, V. Vu (2012)
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non-Gaussian covariance matrices
comparison with Wishart model
partial results
localization results L. Erdos, H.-T. Yau (2009-12)
Lindeberg comparison method T. Tao, V. Vu (2010-11)
Var(NI
)= O(logM)
S. Dallaporta, V. Vu (2011)
P(NI − E(NI ) ≥ t
)≤ C e−ct
δ, t ≥ C logM, 0 < δ ≤ 1
T. Tao, V. Vu (2012)
![Page 133: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/133.jpg)
non-Lipschitz functions f
typically f = 1I , I ⊂ R interval
M∑k=1
f(λNk)
= #{λNk ∈ I
}= NI counting function
Wishart matrices (determinantal structure)
I interval in (a, b)
1√logM
[NI − E(NI )
]→ G Gaussian variable
exponential tail inequalities
P(NI − E(NI ) ≥ t
)≤ C e−ct log(1+t/ logM), t ≥ 0
Var(NI
)= O(logM)
![Page 134: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/134.jpg)
two main questions and objectives
tail inequalities for the spectral measure
P( M∑
k=1
f (λNk ) ≥ t
)
tail inequalities for the extremal eigenvalues
P(λNM ≥ b(ρ) + ε
)
Wishart matrices
more general covariance matrices
![Page 135: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/135.jpg)
two main questions and objectives
tail inequalities for the spectral measure
P( M∑
k=1
f (λNk ) ≥ t
)
tail inequalities for the extremal eigenvalues
P(λNM ≥ b(ρ) + ε
)
Wishart matrices
more general covariance matrices
![Page 136: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/136.jpg)
tail inequalities for the extremal eigenvalues
fluctuations of the largest eigenvalue
M2/3[λNM − b(ρ)
]→ C (ρ)FTW M ∼ ρN
![Page 137: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/137.jpg)
extremal eigenvalues
largest eigenvalue λNM = max1≤k≤M λNk
λNM =λNMN→ b(ρ) =
(1 +√ρ)2
M ∼ ρN
fluctuations around b(ρ)
complex or real Gaussian (Wishart matrices)
M2/3[λNM − b(ρ)
]→ C (ρ)FTW
FTW C. Tracy, H. Widom (1994) distribution
K. Johansson (2000), I. Johnstone (2001)
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tail inequalities for the extremal eigenvalues
fluctuations of the largest eigenvalue
M2/3[λNM − b(ρ)
]→ C (ρ)FTW M ∼ ρN
finite M inequalities
at the (mean)1/3 rate
reflecting the tails of FTW
bounds on Var( λNM)
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tail inequalities for the extremal eigenvalues
fluctuations of the largest eigenvalue
M2/3[λNM − b(ρ)
]→ C (ρ)FTW M ∼ ρN
finite M inequalities
at the (mean)1/3 rate
reflecting the tails of FTW
bounds on Var( λNM)
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tail inequalities for the extremal eigenvalues
fluctuations of the largest eigenvalue
M2/3[λNM − b(ρ)
]→ C (ρ)FTW M ∼ ρN
finite M inequalities
at the (mean)1/3 rate
reflecting the tails of FTW
bounds on Var( λNM)
![Page 141: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/141.jpg)
tail inequalities for the extremal eigenvalues
fluctuations of the largest eigenvalue
M2/3[λNM − b(ρ)
]→ C (ρ)FTW M ∼ ρN
finite M inequalities
at the (mean)1/3 rate
reflecting the tails of FTW
bounds on Var( λNM)
![Page 142: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/142.jpg)
measure concentration tool
(Gaussian) Wishart matrix Y Y t
λNM = max1≤k≤M
λNk = sup|v |=1
|Y v |2
sNM =√λNM Lipschitz of the Gaussian entries Yij
Gaussian concentration
P(sNM ≥ E
(sNM)
+ t)≤ e−M t2/C , t ≥ 0
E(sNM) ∼√
b(ρ)
correct large deviation bounds (t ≥ 1)
![Page 143: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/143.jpg)
measure concentration tool
(Gaussian) Wishart matrix Y Y t
λNM = max1≤k≤M
λNk = sup|v |=1
|Y v |2
sNM =√λNM Lipschitz of the Gaussian entries Yij
Gaussian concentration
P(sNM ≥ E
(sNM)
+ t)≤ e−M t2/C , t ≥ 0
E(sNM) ∼√
b(ρ)
correct large deviation bounds (t ≥ 1)
![Page 144: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/144.jpg)
measure concentration tool
(Gaussian) Wishart matrix Y Y t
λNM = max1≤k≤M
λNk = sup|v |=1
|Y v |2
sNM =√λNM Lipschitz of the Gaussian entries Yij
Gaussian concentration
P(sNM ≥ E
(sNM)
+ t)≤ e−M t2/C , t ≥ 0
E(sNM) ∼√
b(ρ)
correct large deviation bounds (t ≥ 1)
![Page 145: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/145.jpg)
measure concentration tool
(Gaussian) Wishart matrix Y Y t
λNM = max1≤k≤M
λNk = sup|v |=1
|Y v |2
sNM =√λNM Lipschitz of the Gaussian entries Yij
Gaussian concentration
P(sNM ≥ E
(sNM)
+ t)≤ e−M t2/C , t ≥ 0
E(sNM) ∼√
b(ρ)
correct large deviation bounds (t ≥ 1)
![Page 146: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/146.jpg)
measure concentration tool
(Gaussian) Wishart matrix Y Y t
λNM = max1≤k≤M
λNk = sup|v |=1
|Y v |2
sNM =√λNM Lipschitz of the Gaussian entries Yij
Gaussian concentration
P(sNM ≥ E
(sNM)
+ t)≤ e−M t2/C , t ≥ 0
E(sNM) ∼√b(ρ)
correct large deviation bounds (t ≥ 1)
![Page 147: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/147.jpg)
measure concentration tool
(Gaussian) Wishart matrix Y Y t
λNM = max1≤k≤M
λNk = sup|v |=1
|Y v |2
sNM =√λNM Lipschitz of the Gaussian entries Yij
Gaussian concentration
P(sNM ≥ E
(sNM)
+ t)≤ e−M t2/C , t ≥ 0
E(sNM) ∼√b(ρ)
correct large deviation bounds (t ≥ 1)
![Page 148: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/148.jpg)
measure concentration tool
(Gaussian) Wishart matrix Y Y t
λNM = max1≤k≤M
λNk = sup|v |=1
|Y v |2
sNM =√λNM Lipschitz of the Gaussian entries Yij
Gaussian concentration
P(sNM ≥ E
(sNM)
+ t)≤ e−M t2/C , t ≥ 0
E(sNM) ∼√b(ρ)
does not fit the small deviation regime t = s M−2/3
![Page 149: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/149.jpg)
extreme eigenvalues
alternate tools
Riemann-Hilbert analysis (Wishart matrices)
tri-diagonal representations (Wishart and β-ensembles)
moment methods (Wishart and non-Gaussian matrices)
![Page 150: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/150.jpg)
extreme eigenvalues
alternate tools
Riemann-Hilbert analysis (Wishart matrices)
tri-diagonal representations (Wishart and β-ensembles)
moment methods (Wishart and non-Gaussian matrices)
![Page 151: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/151.jpg)
extreme eigenvalues
alternate tools
Riemann-Hilbert analysis (Wishart matrices)
tri-diagonal representations (Wishart and β-ensembles)
moment methods (Wishart and non-Gaussian matrices)
![Page 152: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/152.jpg)
M2/3[λNM − b(ρ)
]→ C (ρ)FTW
P(λNM ≤ b(ρ) + s M−2/3
)→ FTW(C s)
bounds for Wishart matrices
tri-diagonal representation
B. Rider, M. L. (2010)
P(λNM ≥ b(ρ) + ε
)≤ C e−Mε3/2/C , 0 < ε ≤ 1
P(λNM ≤ b(ρ)− ε
)≤ C e−Mε3/C , 0 < ε ≤ b(ρ)
![Page 153: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/153.jpg)
M2/3[λNM − b(ρ)
]→ C (ρ)FTW
P(λNM ≤ b(ρ) + s M−2/3
)→ FTW(C s)
bounds for Wishart matrices
tri-diagonal representation
B. Rider, M. L. (2010)
P(λNM ≥ b(ρ) + ε
)≤ C e−Mε3/2/C , 0 < ε ≤ 1
P(λNM ≤ b(ρ)− ε
)≤ C e−Mε3/C , 0 < ε ≤ b(ρ)
![Page 154: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/154.jpg)
M2/3[λNM − b(ρ)
]→ C (ρ)FTW
P(λNM ≤ b(ρ) + s M−2/3
)→ FTW(C s)
bounds for Wishart matrices
tri-diagonal representation
B. Rider, M. L. (2010)
P(λNM ≥ b(ρ) + ε
)≤ C e−Mε3/2/C , 0 < ε ≤ 1
P(λNM ≤ b(ρ)− ε
)≤ C e−Mε3/C , 0 < ε ≤ b(ρ)
![Page 155: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/155.jpg)
M2/3[λNM − b(ρ)
]→ C (ρ)FTW
P(λNM ≤ b(ρ) + s M−2/3
)→ FTW(C s)
bounds for Wishart matrices
tri-diagonal representation
B. Rider, M. L. (2010)
P(λNM ≥ b(ρ) + ε
)≤ C e−Mε3/2/C , 0 < ε ≤ 1
P(λNM ≤ b(ρ)− ε
)≤ C e−Mε3/C , 0 < ε ≤ b(ρ)
![Page 156: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/156.jpg)
P(λNM ≤ b(ρ) + s M−2/3
)→ FTW(C s)
bounds for Wishart matrices
P(λNM ≥ b(ρ) + ε
)≤ C e−Mε3/2/C , 0 < ε ≤ 1
P(λNM ≤ b(ρ)− ε
)≤ 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( λNM) = O( 1
M4/3
)
![Page 157: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/157.jpg)
P(λNM ≤ b(ρ) + s M−2/3
)→ FTW(C s)
bounds for Wishart matrices
P(λNM ≥ b(ρ) + ε
)≤ C e−Mε3/2/C , 0 < ε ≤ 1
P(λNM ≤ b(ρ)− ε
)≤ 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( λNM) = O( 1
M4/3
)
![Page 158: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/158.jpg)
P(λNM ≤ b(ρ) + s M−2/3
)→ FTW(C s)
bounds for Wishart matrices
P(λNM ≥ b(ρ) + ε
)≤ C e−Mε3/2/C , 0 < ε ≤ 1
P(λNM ≤ b(ρ)− ε
)≤ 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( λNM) = O( 1
M4/3
)
![Page 159: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/159.jpg)
P(λNM ≤ b(ρ) + s M−2/3
)→ FTW(C s)
bounds for Wishart matrices
P(λNM ≥ b(ρ) + ε
)≤ C e−Mε3/2/C , 0 < ε ≤ 1
P(λNM ≤ b(ρ)− ε
)≤ 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( λNM) = O( 1
M4/3
)
![Page 160: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/160.jpg)
M2/3[λNM − b(ρ)
]→ C (ρ)FTW
b(ρ) =(1 +√ρ)2
λNM = λNM/N, M = M(N) ∼ ρN
(√MN)1/3
(√M +
√N)4/3
(λNM − (
√M +
√N)2
)→ FTW
N + 1 ≥ M 0 < ε ≤ 1
P(λNM ≥ (
√M +
√N)2(1 + ε)
)≤ C e
−√MN ε3/2( 1√
ε∧(
MN
)1/4)/C
P(λNM ≤ (
√M +
√N)2(1− ε)
)≤ C e−MN ε3( 1
ε∧(
MN
)1/2)/C
![Page 161: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/161.jpg)
M2/3[λNM − b(ρ)
]→ C (ρ)FTW
b(ρ) =(1 +√ρ)2
λNM = λNM/N, M = M(N) ∼ ρN
(√MN)1/3
(√M +
√N)4/3
(λNM − (
√M +
√N)2
)→ FTW
N + 1 ≥ M 0 < ε ≤ 1
P(λNM ≥ (
√M +
√N)2(1 + ε)
)≤ C e
−√MN ε3/2( 1√
ε∧(
MN
)1/4)/C
P(λNM ≤ (
√M +
√N)2(1− ε)
)≤ C e−MN ε3( 1
ε∧(
MN
)1/2)/C
![Page 162: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/162.jpg)
bi and tri-diagonal representation
B =
χN 0 0 · · · · · · 0
χ(M−1) χN−1 0 0 · · ·...
0 χ(M−2) χN−3 0. . .
...... 0
. . .. . .
. . . 0... · · · . . . χ2 χN−M+2 00 · · · · · · 0 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
![Page 163: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/163.jpg)
bi and tri-diagonal representation
B =
χN 0 0 · · · · · · 0
χ(M−1) χN−1 0 0 · · ·...
0 χ(M−2) χN−3 0. . .
...... 0
. . .. . .
. . . 0... · · · . . . χ2 χN−M+2 00 · · · · · · 0 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
![Page 164: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/164.jpg)
bi and tri-diagonal representation
B =
χN 0 0 · · · · · · 0
χ(M−1) χN−1 0 0 · · ·...
0 χ(M−2) χN−3 0. . .
...... 0
. . .. . .
. . . 0... · · · . . . χ2 χN−M+2 00 · · · · · · 0 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
![Page 165: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/165.jpg)
bi and tri-diagonal representation
B =
χN 0 0 · · · · · · 0
χ(M−1) χN−1 0 0 · · ·...
0 χ(M−2) χN−3 0. . .
...... 0
. . .. . .
. . . 0... · · · . . . χ2 χN−M+2 00 · · · · · · 0 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
![Page 166: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/166.jpg)
bi and tri-diagonal representation
B =
χN 0 0 · · · · · · 0
χ(M−1) χN−1 0 0 · · ·...
0 χ(M−2) χN−3 0. . .
...... 0
. . .. . .
. . . 0... · · · . . . χ2 χN−M+2 00 · · · · · · 0 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
![Page 167: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/167.jpg)
bounds for non-Gaussian entries
moment method E(Tr((YY t)p
))O. Feldheim, S. Sodin (2010)
largest eigenvalue (symmetric, subGaussian entries)
P(λNM ≥ b(ρ) + ε
)≤ C e−M ε3/2/C , 0 < ε ≤ 1
below the mean ?
necessary for variance bounds
![Page 168: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/168.jpg)
bounds for non-Gaussian entries
moment method E(Tr((YY t)p
))O. Feldheim, S. Sodin (2010)
largest eigenvalue (symmetric, subGaussian entries)
P(λNM ≥ b(ρ) + ε
)≤ C e−M ε3/2/C , 0 < ε ≤ 1
below the mean ?
necessary for variance bounds
![Page 169: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/169.jpg)
bounds for non-Gaussian entries
moment method E(Tr((YY t)p
))O. Feldheim, S. Sodin (2010)
largest eigenvalue (symmetric, subGaussian entries)
P(λNM ≥ b(ρ) + ε
)≤ C e−M ε3/2/C , 0 < ε ≤ 1
below the mean ?
necessary for variance bounds
![Page 170: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/170.jpg)
bounds for non-Gaussian entries
moment method E(Tr((YY t)p
))O. Feldheim, S. Sodin (2010)
largest eigenvalue (symmetric, subGaussian entries)
P(λNM ≥ b(ρ) + ε
)≤ C e−M ε3/2/C , 0 < ε ≤ 1
below the mean ?
necessary for variance bounds
![Page 171: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/171.jpg)
variance level
Var( λNM) = O( 1
M4/3
)
S. Dallaporta (2012)
comparison with Wishart model
localization results L. Erdos, H.-T. Yau (2009-12)
Lindeberg comparison method T. Tao, V. Vu (2010-11)
![Page 172: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/172.jpg)
variance level
Var( λNM) = O( 1
M4/3
)
S. Dallaporta (2012)
comparison with Wishart model
localization results L. Erdos, H.-T. Yau (2009-12)
Lindeberg comparison method T. Tao, V. Vu (2010-11)
![Page 173: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/173.jpg)
smallest eigenvalue
soft edge M = M(N) ∼ ρN, ρ < 1
a(ρ) =(1−√ρ
)2
P(λN1 ≤ a(ρ)− ε
)≤ C e−M ε3/2/C , 0 < ε ≤ 1
P(λN1 ≥ a(ρ) + ε
)≤ C e−M ε3/C , 0 < ε ≤ a(ρ)
Wishart matrices B. Rider, M. L. (2010)
![Page 174: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/174.jpg)
smallest eigenvalue
soft edge M = M(N) ∼ ρN, ρ < 1
a(ρ) =(1−√ρ
)2
P(λN1 ≤ a(ρ)− ε
)≤ C e−M ε3/2/C , 0 < ε ≤ 1
P(λN1 ≥ a(ρ) + ε
)≤ C e−M ε3/C , 0 < ε ≤ a(ρ)
Wishart matrices B. Rider, M. L. (2010)
![Page 175: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/175.jpg)
smallest eigenvalue
hard edge M = N, ρ = 1
a(ρ) =(1−√ρ
)2= 0
P(λN1 ≤
ε
N2
)≤ C
√ε+ C e−cN
large families of covariance matrices
M. Rudelson, R. Vershynin (2008-10)
![Page 176: Concentration Inequalities for Random Matrices · Concentration Inequalities for Random Matrices M. Ledoux Institut de Math ematiques de Toulouse, France](https://reader036.vdocuments.net/reader036/viewer/2022062604/5fb6222d11931870b95d04f6/html5/thumbnails/176.jpg)
smallest eigenvalue
hard edge M = N, ρ = 1
a(ρ) =(1−√ρ
)2= 0
P(λN1 ≤
ε
N2
)≤ C
√ε+ C e−cN
large families of covariance matrices
M. Rudelson, R. Vershynin (2008-10)