Random covariance matrices: Universality of local statistics of eigenvalues

By Terence Tao, Van H. Vu. 2009

154 citations recorded by the source corpus.

We study the eigenvalues of the covariance matrix 1/n M∗M of a large rectangular matrix M = Mn,p = (ζij)1≤i≤p;1≤j≤n whose entries are i.i.d. random variables of mean zero, variance one, and having finite C0th moment for some sufficiently large constant C0. The main result of this paper is a Four Moment theorem for i.i.d. covariance matrices (analogous to the Four Moment theorem for Wigner matrices established by the authors in [Acta Math. (2011) Random matrices: Universality of local eigenvalue statistics] (see also [Comm. Math. Phys. 298 (2010) 549–572])). We can use this theorem together with existing results to establish universality of local statistics of eigenvalues under mild conditions. As a byproduct of our arguments, we also extend our previous results on random Hermitian matrices to the case in which the entries have finite C0th moment rather than exponential decay.

Subfields: Applied Mathematics, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/084ab3d5c1b61781d2df4bbd57f5ab9a461dc983

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Citations in the corpus

Universality of local spectral statistics of random matrices (2011, methodology): “ M× N matrix with independent identically distributed entries. The semicircle law is replaced with the Marchenko-Pastur law, but most results listed in this review remain valid. For more details, see [34, 69]. 9 Edge universality Denote by λN is the largest eigenvalue of a generalized Wigner matrix.”
Universality for the largest eigenvalue of sample covariance matrices with general population (2013, unknown): “The delocalization property for the eigenvectors of random matrices was also raised in [12] and improved in the series work [13, 14, 19] for Wigner matrices and extended to sample covariance matrices in [35, 39, 16, 31], which asserts that the coefficients of the unit eigenvectors are in the order o”
Inflation and String Theory (2014, background): “[461,465] — but to simplify the discussion we will continue to omit the associated technicalities.”
RANDOM COVARIANCE MATRICES: UNIVERSALITY OF LOCAL STATISTICS OF EIGENVALUES UP TO THE EDGE (2011, methodology): “The argument draws heavily from those in [13], [12] and [10], thus we only focus on the changes needed to complete the proofs.”
The spectral norm of random inner-product kernel matrices (2015, background): “[50, 30, 55, 26, 45] as well as the recent work of [52, 19] in a nonindependent setting), the model (1) for nonlinear kernels does not have the same independence structure as these models, and it is also not a sum of rank-one updates.”
Universality of covariance matrices (2011, methodology): “3, using the four moment theorem in [38], the subexponential-decay assumption for the matrix entries can be replaced with the existence of a sufficiently high number of moments.”
Bulk universality for generalized Wigner matrices (2010, methodology): “ion that the occurrence of the universality is due to the relaxation to local equilibrium of the DBM. We remark that very recently the results of [31] were also extended to sample covariance matrices [33]. 2 The main input of all these methods [17, 15, 19] and [31, 33] is an estimate of the local de”
Characterizing and correcting for the effect of sensor noise in the dynamic mode decomposition (2014, background): “We direct the interested reader to relevant results in perturbed SVD’s (Stewart 1991, 2006; Bai and Silverstein 2009), random inner product matrices (Cheng and Singer 2013; Tao and Vu 2012), and POD-type operations on noisy data (Epps and Techet 2010; Singer and Wu 2013; Zhao and Singer 2013).”
Random matrix theory in statistics: A review (2014, methodology): “The works of Tao and Vu (2011, 2012a) show that the rate of convergence is at most Oðn 1 log nÞ.”
Random matrices: Universal properties of eigenvectors (2011, background): “One immediate diﬃculty that arises in the discrete setting is that one can now have a non-zero probability that the spectrum is non-simple. However, we have the following gap theorem from [17], [18], [19]: Theorem 5 (Gap theorem). Suppose that Mn is a Wigner random matrix obeying Condition C1 with a”
Geometry of Neural Network Loss Surfaces via Random Matrix Theory (2017, background): “For instance, Tao & Vu (2012) show that the spectrum of Wishart matrices asymptotically approaches the Marcenko-Pastur law regardless of the distribution of the individual entries, so long as they are independent, have mean zero, unit variance, and finite k-th moment for k > 2.”
Edge universality of correlation matrices (2011, methodology): “In previous works mentioned earlier, the authors use a Lindeberg replacement strategy, as in [2, 27].”
Universality of Wigner random matrices: a survey of recent results (2010, methodology): “ 41], but due to certain technical conditions, they were reproved in [96]. The local semicircle law for sample covariance matrices was ﬁrst proved in [46] and a slightly diﬀerent version was given in [98] with some change in the technical assumptions tailored to the application. The four moment cond”