A CLT for Information-Theoretic Statistics of Gram Random Matrices - PowerPoint PPT Presentation

A CLT for Information-Theoretic Statistics of Gram Random Matrices Malika Kharouf Joint work with W.Hachem, J.Najim and J.Silverstein October 12, 2010 Workshop Large Random Matrices and their applications - October 11-13, 2010.

The Model: A Non-Centered Random Matrices Consider a p × n random matrices: 1 √ nX n + A n , Σ n = where, ◮ X n ij , 1 ≤ i ≤ p , 1 ≤ j ≤ n are i.i.d. centered with unit variance and E | X 11 | 16 < ∞ . ◮ A n is a p × n deterministic matrix with uniformly bounded spectral norm.

The Model: Information-Theoretic Statistics of Gram random matrices Linear spectral statistics: p I n ( ρ ) = 1 � � λ ( n ) � log + ρ , i p i =1 where, λ ( n ) , i = 1 , . . . , p are the eigenvalues of the Gram random i matrix Σ n Σ ∗ n and ρ is a nonnegative parameter. Objective: Understanding the asymptotic distribution of the fluctuations of I n ( ρ ), when the dimensions of the matrix Σ n converge to infinity at the same pace and obtain a simple form of the variance.

Plan Motivations: Mutual Information for Multiple Antenna Radio Channels Asymptotic behavior of I n ( ρ ): First-order results Fundamental system of equations Deterministic equivalents Study of the fluctuations Definition of the variance The Central Limit Theorem Outline of the proof of the CLT The approach: REFORM method Main steps of the proof The bias Outline of the proof of the bias term

Motivations: Mutual Information for Multiple Antenna Radio Channels

Multi-user MIMO scheme Figure: MIMO Systems

MIMO System: Mathematical Model The p -dimensional receiver vector r n is given by: r n = Σ n t n + b n , where, ◮ Σ n represents the channel matrix which assumed to be random. ◮ t n is the n -dimensional transmitter vector. ◮ b n is an additive white Gaussian noise with covariance matrix E b n b ∗ n = ρ I p . Performance indicator: The Mutual Information: p I n ( ρ ) = 1 n + ρ I p ) = 1 � p log det (Σ n Σ ∗ log ( λ i + ρ ) p i =1 Asymptotic behavior of I n ( ρ ) when n , p → ∞ at the same rate ?

Asymptotic behavior of I n ( ρ ) : First-order results

First-order results Let f n denotes the ST of µ Σ n Σ ∗ n , the spectral measure of the eigenvalues of Σ n Σ ∗ n . Then, � ∞ I n ( ρ ) = − f n ( − ω ) d ω. ρ Then the asymptotic behavior of I n ( ρ ) is closely linked to the asymptotic behavior of f n as p , n → ∞ with the same pace.

State of the art n → H , H is a deterministic probability measure. ◮ F A n A ∗ Dozier and Silverstein (04): n weakly F Σ n Σ ∗ − − − − − → F , where, F is a deterministic probability measure which the Stieltjes transform is a unique solution of a given coupled equation.

State of the art n → H , H is a deterministic probability measure. ◮ F A n A ∗ Dozier and Silverstein (04): n weakly F Σ n Σ ∗ − − − − − → F , where, F is a deterministic probability measure which the Stieltjes transform is a unique solution of a given coupled equation. ◮ V. L. Girko (91), Hachem-Loubaton-Najim (07) : Look for a deterministic approximation of the Stieltjes transform f n of F Σ n Σ ∗ n . ∃ a p × p deterministic valued function T n ( ρ ) such that: f n ( − ρ ) − 1 a . s p Tr T n ( − ρ ) − − − → n →∞ 0

Fundamental equations Theorem (Girko ’91, Hachem-Loubaton-Najim ’07) The following system of two equations � − 1 △  A n A ∗ δ n ( ρ ) = 1 � = 1 � � 1 + ˜ n δ n ( ρ ) I p + n Tr T n ( ρ )  n Tr ρ   1 + δ n ( ρ )  � − 1 △ A ∗ δ n ( ρ ) = 1 � = 1 n A n ˜ n Tr ˜  n Tr ρ (1 + δ n ( ρ )) I n + T n ( ρ ) ,   1 + ˜  δ n ( ρ ) admits a unique solution ( δ n , ˜ δ n ) in S ( R + ) 2 . Moreover, � � a . s R + f ( λ ) dF Σ n Σ ∗ ∀ f ∈ C B ( R + ) , n ( λ ) − R + f ( λ ) π n ( d λ ) − n →∞ 0 , − − → where π n is the positive measure where δ n is the Stieltjes transform.

First order result: Deterministic equivalents Theorem (Hachem-Loubaton-Najim ’07) � Let V n ( ρ ) = R + log( λ + ρ ) π n ( d λ ) . Then we have: E I n ( ρ ) − V n ( ρ ) − − − − − − − − − − → n → c > 0 0 . n , p →∞ , p Moreover, V n ( ρ ) admits a closed-form expression p � µ 2 � 1 � � n , i � 1 + ˜ V n ( ρ ) = log ρ δ n + 1 + δ n p i =1 + n p log (1 + δ n ) − ρ n p δ n ˜ δ n , where µ n , i are the singular values of the mean matrix A n .

In the non-centered case, the first-order asymptotic study of the mutual information depends mainly on the limiting behavior of the singular values of the mean matrix A n .

Study of the fluctuations

CLT for p ( I n ( ρ ) − V n ( ρ )) In order to study the CLT for p ( I n ( ρ ) − V n ( ρ )) we study separately two quantities: ◮ The random quantity p ( I n ( ρ ) − E I n ( ρ )) from which the fluctuations arise and, ◮ The deterministic quantity p ( E I n ( ρ ) − V n ( ρ )) which yields a bias.

Asymptotic distribution of the fluctuations: Definition of the variance Theorem (Hachem-Kharouf-Najim-Silverstein ’10) 11 , κ = E | X 11 | 4 − 2 − ϑ 2 and let Let ϑ = E X 2 T ¯ n Tr T 2 , ˜ T 2 , γ = 1 γ = 1 γ = 1 n Tr ˜ n Tr T ¯ γ = 1 n Tr ˜ ˜ T , ˜ T . Denote by 2     1 Θ 2 � Tr TAA ∗ T − ρ 2 γ ˜ = − log  1 − γ n    � 1 + ˜ n δ 2 � �   � � 1 � Tr ¯ T ¯ � AA ∗ T � − | ϑ | 2 ρ 2 γ ˜ − log 1 − ϑ γ  � �  � 1 + ˜ � � n δ � � + κρ 2 � � t 2 t 2 ˜ ii jj n 2 i j Then Θ 2 n is well defined.

Some remarks ◮ The variance is the sum of tree terms: the first term would be the same in the Gaussian case. ◮ The variance depends on the singular values of the main matrix as well as on its singular vectors. = X ij e i α for all α ), the second term D ◮ In the circular case ( X ij disappears.

Asymptotic distribution of the fluctuations: The CLT Theorem (Hachem-Kharouf-Najim-Silverstein ’10) The following convergence holds true: p D ( I n ( ρ ) − E I n ( ρ )) − p , n →∞ N (0 , 1) , − − − → Θ n where D stands for convergence in distribution.

Proof of the CLT: The approach REFORM ( RE solvent FOR mula and M artingale). ◮ I n ( ρ ) − E I n ( ρ ) as a sum of increments of martingale. ◮ Identification of the variance.

CLT for martingales Theorem Let Γ ( n ) 1 , . . . , Γ ( n ) be a sequence of increments of martingale with n respect to a given filtration F ( n ) 1 , . . . , F ( n ) n . Assume that there exists a sequence of nonnegative real numbers (Θ 2 n ) n uniformly bounded away from zero and from infinity. Assume that: ◮ n � Γ ( n ) 2 � |F ( n ) P � − Θ 2 − n →∞ 0 . − − → E n j j − 1 j =1 ◮ The Lyapunov’s condition n 1 | 2+ α − � E | Γ ( n ) ∃ α > 0 , n →∞ 0 , − − → holds . j Θ 2(1+ α ) n j =1 j =1 Γ ( n ) � n Then Θ − 1 converges in distribution to N (0 , 1) . n j

Sum of martingale differences We have, n n △ � � I n − E I n = ( E j − E j − 1 ) ( − log(1 + ξ j )) = Γ j , j =1 j =1 where, � � 1 η ∗ n Tr Q j + a ∗ j Q j η j − j Q j a j ξ j = . 1 + 1 n Tr Q j + a ∗ j Q j a j with η j , a j are resp. the jth columns of matrices Σ n and A n , Q j is the resolvent of the matrix Σ j Σ ∗ j and E j stands for the conditional expectation with respect to the σ -algebra F ( n ) = σ ( x 1 , . . . , x j ). j

Sum of the conditional variances Some properties of the function log, n n E j − 1 (( E j − E j − 1 ) log(1 + ξ j )) 2 − � � E j − 1 ( E j ξ j ) 2 P − p , n →∞ 0 − − − → j =1 j =1 where (recall) � � 1 η ∗ n Tr Q j + a ∗ j Q j η j − j Q j a j ξ j = . 1 + 1 n Tr Q j + a ∗ j Q j a j

Study of the sum of conditional variances Standard calculations remain the problem to the study of the asymptotic behavior of the quantities: 1 n Tr ( E j Q n ) 2 j ( E j Q n ) 2 a j , a ∗ and where Q n is the resolvent of Σ n Σ ∗ n matrix.

Outline of the proof A good comprehension of the asymptotic behavior of these terms requires a specific study of bilinear forms of type u ∗ n Q ( ρ ) v n where at least u n or v n is a given column of the deterministic mean matrix A n . If u n and v n are deterministics, Hachem-Loubaton-Najim-Vallet (preprint’10) u ∗ n Q ( ρ ) v n ≈ u ∗ n T ( ρ ) v n

Asymptotic behavior of the bias: Theorem (Hachem-Kharouf-Najim-Silverstein ’10) We have, p ( E I n ( ρ ) − V n ( ρ )) − B n ( ρ ) − p , n →∞ 0 − − − → where, B n ( ρ ) = κ Cte ( ρ, δ, ˜ δ ) κ = E | X 11 | 4 − 2 − ϑ 2 .

Outline of the proof of the bias term The bias term is given by p ( E I n ( ρ ) − V n ( ρ )) χ n ( ρ ) = � ∞ d d ω E log det (Σ n Σ ∗ = n + ω I p ) d ω p ρ � ∞ �� � d − p R + log ( λ + ω ) π n ( d λ ) d ω d ω ρ � ∞ = Tr ( E Q n ( ω ) − T n ( ω )) d ω. ρ Then it remains to study the asymptotic behavior of Tr ( E Q n ( ω ) − T n ( ω )). We prove, Tr ( E Q n ( ω ) − T n ( ω )) − κ Cte ( ρ, δ, ˜ δ ) − n →∞ 0 − − →

Case of a non-centered separable random matrix model