Eigenvectors of some large sample covariance matrix ensembles

By Olivier Ledoit, S. P'ech'e. 2009

255 citations recorded by the source corpus.

We consider sample covariance matrices $${S_N=\frac{1}{p}\Sigma_N^{1/2}X_NX_N^* \Sigma_N^{1/2}}$$ where XN is a N × p real or complex matrix with i.i.d. entries with finite 12th moment and ΣN is a N × N positive definite matrix. In addition we assume that the spectral measure of ΣN almost surely converges to some limiting probability distribution as N → ∞ and p/N → γ > 0. We quantify the relationship between sample and population eigenvectors by studying the asymptotics of functionals of the type $${\frac{1}{N}\text{Tr} ( g(\Sigma_N) (S_N-zI)^{-1}),}$$ where I is the identity matrix, g is a bounded function and z is a complex number. This is then used to compute the asymptotically optimal bias correction for sample eigenvalues, paving the way for a new generation of improved estimators of the covariance matrix and its inverse.

Subfields: Geometry and Topology, Mathematical Physics, Statistics and Probability

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Citations in the corpus

Underspecification Presents Challenges for Credibility in Modern Machine Learning (2020, methodology): “In the underparametrized regime ψ1 < ψ2 the result can be obtained making use of Ledoit and Péché (2011).”
Large Dynamic Covariance Matrices (2019, methodology): “The following theorem, which is equivalent in spirit to EC’s Theorem 1, shows that the loss function in Definition 1 possesses the desired property.”
Quadratic Shrinkage for Large Covariance Matrices (2020, methodology): “All existing ones have been based on a fundamental equation from RMT originally due to Marčenko and Pastur (1967), reformulated by Silverstein (1995), and generalized by Ledoit and Péché (2011).”
High-Dimensional Asymptotics of Prediction: Ridge Regression and Classification (2015, methodology): “We extend the result of Ledoit and P´ech´e (2011), and rely on additional results and ideas from Hachem et al. (2007) and Chen et al. (2011).”
Capturing the learning curves of generic features maps for realistic data sets with a teacher-student model (2021, methodology): “For instance, [60] showed that these assumptions are verified for any data generated as u = Σ u,uω, assuming the components of the vector ω are drawn i.”
The Power of (Non-)Linear Shrinking: A Review and Guide to Covariance Matrix Estimation (2020, methodology): “There are plenty of applications of linear shrinkage to the single-factor matrix or the constant-correlation matrix, going back to the empirical analysis we provided already in Ledoit and Wolf (2003). Most applications are in the context of Markowitz-type portfolio selection and are too numerous to ”
High‐dimensional covariance matrix estimation (2019, methodology): “With respect to minimizing the Frobenius loss ∥∥Σ̂−Σ∥∥2 F , Ledoit and Péché (2011) showed that with a class of rotation-equivariant estimators Σ(D) = PDP, where P = (p1, . . . ,pp) is the matrix of eigenvectors for S and D = diag(d1, . . . , dp) is a diagonal matrix to be determined, the solution i”
Cleaning large correlation matrices: tools from random matrix theory (2016, background): “ctors associated to eigenvalues lying in the bulk of the spectral density when the dimension of the empirical correlation matrix grows to innity. This question has been investigated very recently in [36,37] and we repeat the dierent arguments here. The rst step is to characterize the asymptotic beha”
Random matrix theory in statistics: A review (2014, methodology): “Adifferent approach that deals with projections of sample eigen-subspaces onto population eigen-subspaces has been adopted by Ledoit and Péché (2011) (see Section 4.1).”
Surprises in High-Dimensional Ridgeless Least Squares Interpolation (2019, background): “Technically, [57, 69] apply asymptotic random matrix theory results, such as [43], while we have to take a longer detour to exploit nonasymptotic results established in [41].”
Nonparametric eigenvalue-regularized precision or covariance matrix estimator (2016, methodology): “These four assumptions are very similar to assumptions A1 to A4 in Ledoit and Wolf (2012) and (H1) to (H4) in Ledoit and Péché (2011).”
Nonlinear Shrinkage of the Covariance Matrix for Portfolio Selection: Markowitz Meets Goldilocks (2017, unknown): “B.4 Proof of Proposition 4.2 Theorem 4 of Ledoit and Péché (2011) and the paragraphs immediately above it imply that 1 N N∑ i=1 u′T,iΣTuT,i a.s.−→ ∫ d∗(x) dF (x) .”