Universality of covariance matrices
By Natesh S. Pillai, Jun Yin. 2011
178 citations recorded by the source corpus.
In this paper we prove the universality of covariance matrices of the form $H_{N\times N}={X}^{\dagger}X$ where $X$ is an ${M\times N}$ rectangular matrix with independent real valued entries $x_{ij}$ satisfying $\mathbb{E}x_{ij}=0$ and $\mathbb{E}x^2_{ij}={\frac{1}{M}}$, $N$, $M\to \infty$. Furthermore it is assumed that these entries have sub-exponential tails or sufficiently high number of moments. We will study the asymptotics in the regime $N/M=d_N\in(0,\infty),\lim_{N\to\infty}d_N\neq0,\infty$. Our main result is the edge universality of the sample covariance matrix at both edges of the spectrum. In the case $\lim_{N\to\infty}d_N=1$, we only focus on the largest eigenvalue. Our proof is based on a novel version of the Green function comparison theorem for data matrices with dependent entries. En route to proving edge universality, we establish that the Stieltjes transform of the empirical eigenvalue distribution of $H$ is given by the Marcenko-Pastur law uniformly up to the edges of the spectrum with an error of order $(N\eta)^{-1}$ where $\eta$ is the imaginary part of the spectral parameter in the Stieltjes transform. Combining these results with existing techniques we also show bulk universality of covariance matrices. All our results hold for both real and complex valued entries.
Subfields: Mathematical Physics, Statistics and Probability, Statistical Learning Theory
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Citations in the corpus
- The local semicircle law for a general class of random matrices (2012, background): “dierent proof (with a better bound on the constants) was given in [19]. A conceptually streamlined version of the original proof was extended to sparse matrices [6] and to sample covariance matrices [26]. Finally, an extensive analysis in [9] treated the uctuation averaging of general polynomials of”
- Spiked separable covariance matrices and principal components (2019, background): “Gaussian), whereas the edge universality for X with generally distributed entries was proved later in [48].”
- Statistical inference for principal components of spiked covariance matrices (2020, background): “[72] N.”
- A necessary and sufficient condition for edge universality at the largest singular values of covariance matrices (2016, methodology): “This goal can be accomplished with a Green function comparison method, which has been applied successfully in proving the universality of covariance matrices [29, 30].”
- A Dynamical Approach to Random Matrix Theory (2017, background): “A conceptually streamlined version of the original proof was extended to sparse matrices [56] and to sample covariance matrices [113].”
- Tracy-Widom law for the extreme eigenvalues of sample correlation matrices (2011, unknown): “For α = 1, we have p∑ k=1 1 |λk − z| = p ∫ 1 |x− z|dFp(x).”
- Averaging Fluctuations in Resolvents of Random Band Matrices (2012, methodology): “ of this generalization. Remark 4.14. In [17, Lemma 5.2], a uctuation averaging theorem of the form 1 N X i Q i X(i) k;l h ikG (i) kl h li ˚ 2 (4.8) was proved. This result was further generalized in [5,16,18]. The estimate (4.8) also follows from Theorem 4.8. To see this, we use Schur’s formula (3.”
- Tracy-Widom Distribution for the Largest Eigenvalue of Real Sample Covariance Matrices with General Population (2014, methodology): “The arguments given here are small modifications of the methods presented in [19, 34, 28].”
- Lectures on the local semicircle law for Wigner matrices (2016, methodology): “We remark that a generalization of the methods of [42, 65] for sample covariance matrices may also be used to obtain local laws for non-Hermitian matrices with independent entries, using Girko’s Hermitization trick [46]; see [23, 24, 80] for more details.”
- Universality for the largest eigenvalue of sample covariance matrices with general population (2013, background): “rsality. Specifically, for sample covariance matrices in the null case, the Tracy–Widom law has been established for W under very general assumptions on the distribution of X. The readers can refer to [43, 46, 49, 56] for some representative developments on this topic. For generally distributed spike”