The eigenvalues and eigenvectors of finite, low rank perturbations of large random matrices

By Florent Benaych-Georges, Raj Rao Nadakuditi. 2009

587 citations recorded by the source corpus.

We consider the eigenvalues and eigenvectors of finite, low rank perturbations of random matrices. Specifically, we prove almost sure convergence of the extreme eigenvalues and appropriate projections of the corresponding eigenvectors of the perturbed matrix for additive and multiplicative perturbation models. The limiting non-random value is shown to depend explicitly on the limiting eigenvalue distribution of the unperturbed random matrix and the assumed perturbation model via integral transforms that correspond to very well known objects in free probability theory that linearize non-commutative free additive and multiplicative convolution. Furthermore, we uncover a phase transition phenomenon whereby the large matrix limit of the extreme eigenvalues of the perturbed matrix differs from that of the original matrix if and only if the eigenvalues of the perturbing matrix are above a certain critical threshold. Square root decay of the eigenvalue density at the edge is sufficient to ensure that this threshold is finite. This critical threshold is intimately related to the same aforementioned integral transforms and our proof techniques bring this connection and the origin of the phase transition into focus. Consequently, our results extend the class of `spiked' random matrix models about which such predictions (called the BBP phase transition) can be made well beyond the Wigner, Wishart and Jacobi random ensembles found in the literature. We examine the impact of this eigenvalue phase transition on the associated eigenvectors and observe an analogous phase transition in the eigenvectors. Various extensions of our results to the problem of non-extreme eigenvalues are discussed.

Subfields: Discrete Mathematics and Combinatorics, Mathematical Physics, Statistics and Probability, Statistical Learning Theory

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Citations in the corpus

Limits of spiked random matrices I (2010, methodology): “A broad generalization of the results on a.s. limits is developed by Benaych-Georges and Nadakuditi [15] and dubbed “spiked free probability theory”.”
Emergent properties of the local geometry of neural loss landscapes (2019, background): “This bulk plus outlier spectral structure is extremely well understood in a wide array of simpler random matrix models [21, 22].”
Outlier Eigenvalues for Deformed I.I.D. Random Matrices (2014, methodology): “As in previous works such as [9, 10], we will use the identity, if P,Q> ∈MN,r(C), det(IN + PQ) = det(Ir +QP ).”
Factors That Fit the Time Series and Cross-Section of Stock Returns (2018, background): “Lettau and Pelger (2019) generalize the spiked covariance models from random matrix theory and properties in Onatski (2012) and Benaych-Georges and Nadakuditi (2011) to analyze the asymptotic behavior of the RP-PCA estimator for weak factors.”
The singular values and vectors of low rank perturbations of large rectangular random matrices (2011, methodology): “In the case where the θ i ’s are not pairwise distinct, an approximation approach allows to conclude (proceed for example as in Section 6.2.3 of [17], using [32, Cor. 7.3.8 (b)] instead of [32, Cor. 6.3.8]).”
Complex diffusion-weighted image estimation via matrix recovery under general noise models (2018, background): “shrinkage looks for the best possible asymptotic signal predictor obtained by only manipulating the singular values of convenient matrix arrangements while corresponding singular vector orientations (Benaych-Georges and Nadakuditi, 2011) are left unaltered. This model does not exploit other reasonab”
Estimating latent asset-pricing factors (2018, unknown): “(13)The properties of weak factor models based on covariances have already been studied in Onatski (2012), Paul (2007) and Benaych-Georges and Nadakuditi (2011). We replicate those results applied to our setup.”
Speeding up Permutation Testing in Neuroimaging (2013, methodology): “Thus, rather than analyzing the singular value spectrum of P directly, we can analyze the eigenvalues of PP using a recent result from [24].”
On sample eigenvalues in a generalized spiked population model (2008, unknown): “Throughout the paper and for any Hermitian matrix A, we order its eigenvalues in an descending order as λA1 ≥ λA2 ≥ · · · .”
The Dynamics of Learning: A Random Matrix Approach (2018, methodology): “We refer the reader to (Benaych-Georges & Nadakuditi, 2011) for an introduction.”
Identifying the number of factors from singular values of a large sample auto-covariance matrix (2014, methodology): “For some recent accounts on this theory, we refer to Johnstone [15], Baik and Silverstein [6], Bai and Yao [4], Benaych-Georges and Nadakuditi [7], Passemier and Yao [19] and the references therein.”