ASYMPTOTICS OF SAMPLE EIGENSTRUCTURE FOR A LARGE DIMENSIONAL SPIKED COVARIANCE MODEL

By Debashis Paul. 2007

794 citations recorded by the source corpus.

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Statistics and Probability

Citations in the corpus

The Full Spectrum of Deep Net Hessians At Scale: Dynamics with Sample Size (2018, background): “Much is known about its properties [2, 3, 13] and it is therefore of interest to know whether it indeed models the spectrum of the Hessian faithfully.”
Bi-cross-validation for factor analysis (2015, neutral): “See [49, 9, 51] for statements of this result.”
Determining the number of components in a factor model from limited noisy data (2008, background): “This issue has also received considerable attention in recent years [2,29,25].”
The largest eigenvalues of finite rank deformation of large Wigner matrices: Convergence and nonuniversality of the fluctuations. (2007, methodology): “est eigenvalue(s) of Σp. The approach of [8] does not extend to the real Gaussian setting and the whole analogue of their result is still an open question. Nevertheless, Paul was able to establish in [25] the Gaussian ﬂuctuations of the largest eigenvalue of the real Gaussian matrix SN when the larg”
Principal Component Analysis of High-Frequency Data (2016, positive): “In this scenario, Johnstone and Lu (2009) and Paul (2007) showed that the estimated eigenvector corresponding to the largest eigenvalue is inconsistent unless the sample size grows at a rate faster than the one at which the cross-sectional dimension increases.”
Limiting laws for divergent spiked eigenvalues and largest nonspiked eigenvalue of sample covariance matrices (2017, background): “Paul [36] showed that if p/n → γ ∈ (0, 1) as n → ∞, and the largest eigenvalue μ1 of Σ satisfies μ1 ≤ 1 + √ γ, then the leading sample principal eigenvector v̂1 is asymptotically almost surely orthogonal to the leading population eigenvector v1, i.”
Normal approximation and concentration of spectral projectors of sample covariance (2015, background): “It includes, in particular, high-dimensional spiked covariance models (see [9, 10, 19]) in which”
Fundamental limits of detection in the spiked Wigner model (2018, background): “Estimation using the top eigenvector undergoes the same transition, where it is known to “lose track” of the spike below the spectral threshold (Paul, 2007; Nadler, 2008; Johnstone and Lu, 2009; Benaych-Georges and Nadakuditi, 2011).”
Canonical correlation coefficients of high-dimensional Gaussian vectors: Finite rank case (2017, background): “Similar problems have been studied widely for various random matrix models, not trying to be comprehensive, we refer to the spiked sample covariance matrices [34, 6, 44, 4, 5, 22], the deformed Wigner matrices [15, 45, 21, 16, 40], the deformed unitarily invariant matrices [11, 38], and some other d”
Concentration of measure and spectra of random matrices: Applications to correlation matrices, elliptical distributions and beyond (2009, background): “Since this result, there has been a flurry of activity, especially in recent years, concerning the behavior of the largest eigenvalue of sample covariance matrices [21, 50], their fluctuation behavior in the null case [14, 19, 28, 29] and under alternatives [6, 16, 40], as well as fluctuation result”
Asymptotics and Concentration Bounds for Bilinear Forms of Spectral Projectors of Sample Covariance (2014, methodology): “Paul [25] established that θ̂B,j ‖θ̂B,j‖ is uniformly distributed in the unit sphere Sp−m−1 and is independent of ‖θ̂B,j‖.”
The eigenvalues and eigenvectors of finite, low rank perturbations of large random matrices (2009, background): “This trick cannot be applied for Wishart matrices since Wishart matrices do not exhibit the symmetries of Wigner matrices.”
Sparse PCA: Optimal rates and adaptive estimation (2012, methodology): “This estimate works because σ1(S) − 2 is an over-estimate of λ with high probability [39, 43], since the noise variance here is two.”
Spectrum estimation for large dimensional covariance matrices using random matrix theory (2006, background): “It is known that in “large n, large p” asymptotics, the eigenvectors of the sample covariance matrix are not consistent estimators of the population eigenvectors (see [33]), even in the most favorable cases.”
Implicit Self-Regularization in Deep Neural Networks: Evidence from Random Matrix Theory and Implications for Learning (2018, methodology): “In statistics, this corresponds to the Spiked Covariance model [68, 109, 69].”