Eigenvalues of large sample covariance matrices of spiked population models

By Jinho Baik, J. Silverstein. 2004

805 citations recorded by the source corpus.

We consider a spiked population model, proposed by Johnstone, in which all the population eigenvalues are one except for a few fixed eigenvalues. The question is to determine how the sample eigenvalues depend on the non-unit population ones when both sample size and population size become large. This paper completely determines the almost sure limits of the sample eigenvalues in a spiked model for a general class of samples.

Subfields: Condensed Matter Physics, Mathematical Physics, Statistics and Probability, Probabilistic & Bayesian Methods

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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.”
The spectral norm of random inner-product kernel matrices (2015, background): “However, in the “subcritical” regime λ < λ ∗, |v̂T v| → 0 almost surely, λmax(Σ̂) cannot distinguish the null and spiked models, and furthermore no test using only the eigenvalues of Σ̂ can distinguish the models with probability approaching one [32, 3, 4, 44, 42, 43].”
The high-dimension, low-sample-size geometric representation holds under mild conditions (2007, methodology): “Some recent high-dimension, low-sample-size asymptotic studies on sample covariance matrices, including Baik et al. (2005), Baik & Silverstein (2006) and a University of California, Davis technical report by D. Paul, found that, when the ratio d/n goes to a constant, the sample eigenvalues behave…”
The Largest Eigenvalue of Rank One Deformation of Large Wigner Matrices (2006, methodology): “4, the almost sure limit of the largest eigenvalues of any complex or real Deformed Wigner model (MN )N is considered, using an approach due to [ 3 ].”
Statistical eigen-inference from large Wishart matrices (2007, background): “At the same time, however, the results in Baik and Silverstein (2006), Paul (2005), Baik, Ben Arous and Péché (2005) suggest that if the practitioner has reason to believe that the population eigenvalues can be split into several clusters about ai ± ai √ p/n, then the use of the model in (1.2)…”
Rank 1 real Wishart spiked model (2011, background): “These non-trivial eigenvalues in Σ will then be responsible for the spikes that appear in the eigenvalue distribution of the sample covariance matrix [11].”
The singular values and vectors of low rank perturbations of large rectangular random matrices (2011, result): “Our results are in a certain sense very general (in terms of possible distributions for the noise model X) and recover as a special case results found in the literature for the eigenvalues [8, 9] and eigenvectors [34, 46, 44] of X̃X̃∗ in the setting where X in (1) is Gaussian.”
A test metric for assessing single-cell RNA-seq batch correction (2018, background): “However, as the number of features (genes) increases, the largest and smallest eigenvalues of the sample covariance matrix converge 42 where W is the index set of all N top PCs that are significantly correlated with the batch covariate (Supplementary Note 2).”
Outliers in the spectrum of iid matrices with bounded rank perturbations (2010, methodology): “on). See also [21], [13] for further results in the Wigner case, whose methods are close to those used here, [35] for a treatment of the GUE case, [11] for a treatment of the LUE case, and [5], [12], [7] for a treatment of the covariance matrix case. Interestingly, in the Wigner case the outlier eig”
Estimating Number of Factors by Adjusted Eigenvalues Thresholding (2019, neutral): “1 of Baik and Silverstein (2006), we have (1 − )−1λK+1(Sn) = (1 + √ ρ)2 + oa.”
Central limit theorems for eigenvalues in a spiked population model (2008, background): “Therefore, Baik and Silverstein [6] proves that for each k ∈ {1, . . . ,Mb} and sk−1 j ≤ sk (largest eigenvalues) or k ∈ {1, . . . ,Ma} and p− tk j ≤ p− tk−1 (smallest eigenvalues), λn,j → φ(αk) = αk + yαk αk − 1 , almost surely.”
High-dimensional analysis of semidefinite relaxations for sparse principal components (2008, result): “Form the associated order statistics D(1) ≤ D(2) ≤ · · · ≤ D(p−1) ≤ D(p),”
Tracy-Widom Distribution for the Largest Eigenvalue of Real Sample Covariance Matrices with General Population (2014, result): “to the lack of an analogue of the Harish-Chandra-Itzykson-Zuber integral for the orthogonal group. Relying on quite diﬀerent methods, almost sure convergence of the largest eigenvalues was derived in [5] and Tracy-Widom ﬂuctuations of the largest eigenvalue of spiked population models were obtained ”
Statistical Significance of Clustering for High-Dimension, Low–Sample Size Data (2008, background): “Baik and Silverstein (2004) studied the asymptotic behavior of sample eigenvalues when both d and n go to inﬁnity as d/n converges to r .”
The Isotropic Semicircle Law and Deformation of Wigner Matrices (2011, methodology): “Similar identities were also used in [1, 2, 4] for deformed covariance matrices.”
Free Convolution with a Semicircular Distribution and Eigenvalues of Spiked Deformations of Wigner Matrices (2010, neutral): “F´evrier ¶”
On sample eigenvalues in a generalized spiked population model (2008, background): “Keywords and phrases: Sample covariance matrices, Spiked population model, Central limit theorems, Largest eigenvalue, Extreme eigenvalues.”
PCA CONSISTENCY IN HIGH DIMENSION, LOW SAMPLE SIZE CONTEXT (2009, background): “In recent years, substantial work has been done on the asymptotic behavior of eigenvalues of the sample covariance matrix in the limit as d→∞, see Baik, Ben Arous and Péché [2], Johnstone [11] and Paul [16] for Gaussian assumptions and Baik and Silverstein [3] for non-Gaussian results when d and n”