On the distribution of the largest eigenvalue in principal components analysis

By Iain M. Johnstone. 2001

2,278 citations recorded by the source corpus.

Subfields: Computational Mathematics, Computational Theory and Mathematics, Statistics and Probability, Probabilistic & Bayesian Methods

How researchers see this paper

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

In Computational Mathematics

In Computational Theory and Mathematics

In Statistics and Probability

In Probabilistic & Bayesian Methods

Citations in the corpus

The Full Spectrum of Deep Net Hessians At Scale: Dynamics with Sample Size (2018, background): “Certain observed properties of deepnet Hessian spectra are qualitatively similar to those seen in the theory of covariane estimation in modern high-dimensional statistics, where they arise under the so-called spiked covariance model [7].”
Ligand biological activity predicted by cleaning positive and negative chemical correlations (2019, background): “2 describes the typical behavior of random matrices rather than extreme value distributions (33).”
Introduction to the non-asymptotic analysis of random matrices (2010, neutral): “[38].”
Noisy matrix decomposition via convex relaxation: Optimal rates in high dimensions (2011, methodology): “slogd n +c1 α2s d2 . We note that the condition n ≥ d is necessary to obtain consistent estimates in factor analysis models, even in the case with Γ⋆ = Id×d where PCA is possible (e.g., see Johnstone [15]). Again, the terms in the bound have a natural interpretation: since a matrix of rank r in d di”
Sparse Principal Component Analysis and Iterative Thresholding (2011, methodology): “Since there are ¯ m spikes ( λ 21 , . . . , λ 2¯ m ) in the spectrum of Σ, (1.2) has been called the spiked covariance model in the literature [12].”
Positive Definite $\ell_1$ Penalized Estimation of Large Covariance Matrices (2012, background): “The usual sample covariance matrix is optimal in the classical setting with large samples and ﬁxed low dimensions (Anderson, 1984), but it performs very poorly in the high-dimensional setting (Johnstone, 2001).”
Edge universality of correlation matrices (2011, methodology): “Therefore, it is often recommended first to standardize the matrix entries and then perform PCA on the resulting correlation matrix [17].”
Randomized numerical linear algebra: Foundations and algorithms (2020, methodology): “For example, let us consider a variant of the spiked covariance model that is common in statistics applications (Johnstone 2001).”
Spiked separable covariance matrices and principal components (2019, methodology): “The Tracy-Widom distribution of the extremal eigenvalue was first proved in [26] for sample covariance matrices with r A “ Ip and Gaussian X (i.”
Optimal estimation of a large-dimensional covariance matrix under Stein’s loss (2018, neutral): “To distinguish our model from Johnstone’s (2001) “spike” model, where the largest eigenvalues remains bounded, we call the eigenvalues that shoot up to infinity “arrows”.”