Exact separation of eigenvalues of large information plus noise - PowerPoint PPT Presentation

Exact separation of eigenvalues of large information plus noise complex Gaussian models Philippe Loubaton, Pascal Vallet Universit´ e de Paris-Est / Marne la Vall´ ee, LIGM 11/10/2010

Behaviour of the eigenvalue distribution of ˆ Exact separation of the eigenvalues of ˆ Problem statement. R N . R N . Conclusio Plan Problem statement. 1 Behaviour of the eigenvalue distribution of ˆ R N . 2 Exact separation of the eigenvalues of ˆ R N . 3 Conclusion 4

Behaviour of the eigenvalue distribution of ˆ Exact separation of the eigenvalues of ˆ Problem statement. R N . R N . Conclusio Plan Problem statement. 1 Behaviour of the eigenvalue distribution of ˆ R N . 2 Exact separation of the eigenvalues of ˆ R N . 3 Conclusion 4

Behaviour of the eigenvalue distribution of ˆ Exact separation of the eigenvalues of ˆ Problem statement. R N . R N . Conclusio The information plus noise model Introduced in Dozier-Silverstein-2007. M ( N ) × N matrix Σ N Σ N = B N + σ W N B N deterministic matrix sup N � B N � < + ∞ W N zero mean complex Gaussian i.i.d. matrix E | W N , i , j | 2 = 1 N

Behaviour of the eigenvalue distribution of ˆ Exact separation of the eigenvalues of ˆ Problem statement. R N . R N . Conclusio Problem statement Empirical covariance matrix ˆ R N = Σ N Σ ∗ N ( M , N ) → + ∞ , c N = M N → c < 1 Prove the ”Exact Separation” of the eigenvalues of ˆ R N Property introduced by Bai and Silverstein 1999 in the context of zero mean possibly non Gaussian correlated random matrices

Behaviour of the eigenvalue distribution of ˆ Exact separation of the eigenvalues of ˆ Problem statement. R N . R N . Conclusio Numerical illustration (I). σ 2 = 2 N 0 with multiplicity M Eigenvalues of B N B ∗ 2 , 5 with multiplicity M 2 c N = M N , c N = 0 . 2 Representation of histograms of the eigenvalues of ˆ R N

Behaviour of the eigenvalue distribution of ˆ Exact separation of the eigenvalues of ˆ Problem statement. R N . R N . Conclusio Numerical illustration (II). c = M N = 0 . 2

Behaviour of the eigenvalue distribution of ˆ Exact separation of the eigenvalues of ˆ Problem statement. R N . R N . Conclusio Motivation See the talk of P. Vallet tomorrow Rank ( B N ) = K ( N ) < M Π N orthogonal projection matrix on ( Range ( B N )) ⊥ Subspace estimation methods. Estimate consistently a ∗ N Π N a N from Σ N Needs to evaluate the location of the eigenvalues of ˆ R N

Behaviour of the eigenvalue distribution of ˆ Exact separation of the eigenvalues of ˆ Problem statement. R N . R N . Conclusio Plan Problem statement. 1 Behaviour of the eigenvalue distribution of ˆ R N . 2 Exact separation of the eigenvalues of ˆ R N . 3 Conclusion 4

Behaviour of the eigenvalue distribution of ˆ Exact separation of the eigenvalues of ˆ Problem statement. R N . R N . Conclusio The ”asymptotic” limit eigenvalue distribution µ N Notation N → + ∞ stands for ( M , N ) → + ∞ , c N = M N → c < 1 (ˆ λ k , N ) k = 1 ,..., M eigenvalues of ˆ R N , ( λ k , N ) k = 1 ,..., M eigenvalues of B N B ∗ N , arranged in decreasing order Rank ( B N ) = K ( N ) < M , λ K + 1 , N = . . . = λ M , N = 0 Dozier-Silverstein 2007 : It exists a deterministic probability measure µ N carried by R + such that � M 1 k = 1 δ ( λ − ˆ λ k , N ) − µ N → 0 weakly almost surely M

Behaviour of the eigenvalue distribution of ˆ Exact separation of the eigenvalues of ˆ Problem statement. R N . R N . Conclusio How to characterize µ N es transform m N ( z ) of µ N The Stieltj` µ N ( d λ ) m N ( z ) = defined on C − R + � λ − z R + m N ( z ) is solution of the equation m N ( z ) 1 + σ 2 c N m N ( z ) = f N ( w N ( z )) w N ( z ) = z ( 1 + σ 2 c N m N ( z )) 2 − σ 2 ( 1 − c N )( 1 + σ 2 c N m N ( z )) � M f N ( w ) = 1 N − w I M ) − 1 = 1 M Trace ( B N B ∗ 1 M k = 1 λ k , N − w

Behaviour of the eigenvalue distribution of ˆ Exact separation of the eigenvalues of ˆ Problem statement. R N . R N . Conclusio Properties of µ N , c N = M N < 1 S N support of µ N Dozier-Silverstein-2007 For each x ∈ R , lim z → x , z ∈ C + m N ( z ) = m N ( x ) exists x → m N ( x ) continuous on R , continuously differentiable on R \ ∂ S N µ N ( d λ ) absolutely continuous, density 1 π Im ( m N ( x ))

Behaviour of the eigenvalue distribution of ˆ Exact separation of the eigenvalues of ˆ Problem statement. R N . R N . Conclusio Characterization of S N . Reformulation of D-S 2007 in Vallet-Loubaton-Mestre-2009 Function φ N ( w ) defined on R by φ N ( w ) = w ( 1 − σ 2 c N f N ( w )) 2 + σ 2 ( 1 − c N )( 1 − σ 2 c N f N ( w )) φ N has 2 Q positive extrema with preimages w ( N ) 1 , − < w ( N ) 1 , + < . . . w ( N ) Q , − < w ( N ) Q , + . These extrema verify x ( N ) 1 , − < x ( N ) 1 , + < . . . x ( N ) Q , − < x ( N ) Q , + . S N = [ x ( N ) 1 , − , x ( N ) 1 , + ] ∪ . . . [ x ( N ) Q , − , x ( N ) Q , + ] Each eigenvalue λ l , N of B N B ∗ N belongs to an interval ( w ( N ) k , − , w ( N ) k , + )

Behaviour of the eigenvalue distribution of ˆ Exact separation of the eigenvalues of ˆ Problem statement. R N . R N . Conclusio φ ( w ) Support S x + 3 x − 3 x + 2 x − 2 x + 1 x − 1 w − w 1 λ 4 λ 3 λ 2 λ 1 w + w + 2 1 w − w − w + 2 3 3

Behaviour of the eigenvalue distribution of ˆ Exact separation of the eigenvalues of ˆ Problem statement. R N . R N . Conclusio Some definitions Each interval [ x ( N ) q , − , x ( N ) q , + ] is called a cluster An eigenvalue λ l , N of B N B ∗ N is said to be associated to cluster [ x ( N ) q , − , x ( N ) q , + ] if λ l , N ∈ ( w ( N ) q , − , w ( N ) q , + ) 2 eigenvalues of B N B ∗ N are said to be separated if they are associated to different clusters

Behaviour of the eigenvalue distribution of ˆ Exact separation of the eigenvalues of ˆ Problem statement. R N . R N . Conclusio Some useful properties of w N ( x ) w N ( x ) = x ( 1 + σ 2 c N m N ( x )) 2 − σ 2 ( 1 − c N )( 1 + σ 2 c N m N ( x )) . φ N ( w N ( x )) = x for each x Int ( S N ) = { x , Im ( w N ( x )) > 0 } w N ( x ) is real and increasing on each component of S c N w N ( x − q , N ) = w − q , N , w N ( x + q , N ) = w + q , N w N ( x ) is continuous on R and continuously differentiable on R \ ∂ S N | w N ( x ) | ≃ q , N | 1 / 2 if x ≃ x − , + ′ 1 q , N | x − x − , +

Behaviour of the eigenvalue distribution of ˆ Exact separation of the eigenvalues of ˆ Problem statement. R N . R N . Conclusio Contours associated to function x → w N ( x ) (I) Illustration 2 clusters. Im{ w ( x )} w ( x − 2 ) = w − w ( x + 2 ) = w + 2 2 0 λ 4 λ 3 λ 2 λ 1 Re{ w ( x )} w ( x + 1 ) = w + w ( x − 1 ) = w − 1 1

Behaviour of the eigenvalue distribution of ˆ Exact separation of the eigenvalues of ˆ Problem statement. R N . R N . Conclusio Contours associated to function x → w N ( x ) (II) C q = { w N ( x ) , x ∈ [ x − q , N , x + q , N ] } ∪ { w N ( x ) ∗ , x ∈ [ x − q , N , x + q , N ] } Encloses the eigenvalues of B N B ∗ N associated to cluster [ x − q , N , x + q , N ] Continuously differentiable path (except at x − q , N , x + q , N where | w N ( x ) | ≃ ′ 1 q , N | 1 / 2 ) | x − x − , + g ( w ) continuous in a neighborhood of C q , g ( w ∗ ) = g ( w ) ∗ � x + � q , N � � g ( w ) dw = 2 i g ( w N ( x )) w N ( x ) dx Im ′ x − C − q q , N

Behaviour of the eigenvalue distribution of ˆ Exact separation of the eigenvalues of ˆ Problem statement. R N . R N . Conclusio Plan Problem statement. 1 Behaviour of the eigenvalue distribution of ˆ R N . 2 Exact separation of the eigenvalues of ˆ R N . 3 Conclusion 4

Behaviour of the eigenvalue distribution of ˆ Exact separation of the eigenvalues of ˆ Problem statement. R N . R N . Conclusio The results. Theorem 1 Let [ a , b ] such that ] a − ǫ, b + ǫ [ ⊂ ( S N ) c for each N > N 0 . Then, almost surely, for N large enough, none of the eigenvalues of R N appears in [ a , b ] . ˆ Theorem 2 Let [ a , b ] such that ] a − ǫ, b + ǫ [ ⊂ ( S N ) c for each N > N 0 . Then, almost surely, for N large enough, card { k : ˆ λ k , N < a } card { k : λ k , N < w N ( a ) } = card { k : ˆ λ k , N > b } card { k : λ k , N > w N ( b ) } =

Behaviour of the eigenvalue distribution of ˆ Exact separation of the eigenvalues of ˆ Problem statement. R N . R N . Conclusio Existing related results. Bai and Silverstein 1998 in the context of the model Y = HW , W possibly non Gaussian Capitaine, Donati-Martin, and Feral 2009 in the context of the deformed Wigner model Y = A + X , X Gaussian i.i.d. Wigner matrix (or entries verifying the Poincar´ e-Nash inequality), A deterministic hermitian matrix with constant rank. No previous result in the context of the Information plus Noise model