Abstract
We study the spectral norm of random lifts of matrices. Given an n×n symmetric matrix A, and a centered distribution π on k×k(k≥2) symmetric matrices with spectral norm at most 1, let the matrix random lift A(k,π) be the random symmetric kn×kn matrix (AijXij)1≤i<j≤n, where Xij are independent samples from π. We prove that E‖A(k,π)‖≲max i ∑j Aij2+maxij|Aij| log(kn). This result can be viewed as an extension of existing spectral bounds on random matrices with independent entries, providing further instances where the multiplicative logn factor in the Non-Commutative Khintchine inequality can be removed. As a direct application of our result, we prove an upper bound of 2(1+ϵ) Δ+O( log(kn)) on the new eigenvalues for random k-lifts of a fixed G=(V,E) with |V|=n and maximum degree Δ, compared to the previous result of O( Δlog(kn)) by Oliveira [Oli10a] and the recent breakthrough by Bordenave and Collins [BC19] which gives 2 Δ−1+o(1) as k→∞ for Δ-regular graph G.
Highlights
This result can be viewed as an extension of existing spectral bounds on random mat√rices with independent entries, providing further instances where the multiplicative log n factor in the Non-Commutative Khintchine inequality can be removed. √ As a direct application of our result, we prove an upper bound of 2(1 + ) ∆ + O( log(kn)) on the new eigenvalues for random k-lifts of a fixed G = (V, E) with |V | = n and maximum degree ∆, compared to the previous result of O( ∆ log(kn)) by Oliveira [O√li10a] and the recent breakthrough by Bordenave and Collins [BC19] which gives 2 ∆ − 1 + o(1) as k → ∞ for ∆-regular graph G
The Non-Commutative Khintchine (NCK) inequality and other phenomena of matrix concentration have been proven under various settings and extensively studied in [Oli10b, Tro12, Tro15]
(v Aiv)2 i=1 max |bij|, ij σ∗ defined in (1.6) is consistent with the quantity defined in (1.5) in that they differ only by a multiplicative constant, and the proposed improvement (1.7) is true in the case of random matrices with independent entries due to [BvH16]. We show another class of examples, in which we improve the bound given by the NCK inequality (1.2) by replacing the multiplicative log factor with an additive factor, as is the case in the conjectured bound (1.7)
Summary
Piquard and Pisier [Pis03], is one of the simplest tools for understanding the spectrum of matrix series, namely. N ) are n × n real symmetric matrices and γi N ) are i.i.d. random variables, usually assumed gaussian or Rademacher. AN be n × n symmetric matrices and γ1, γ2, . ΓN be i.i.d. N (0, 1) random variables, . The NCK inequality and other phenomena of matrix concentration have been proven under various settings and extensively studied in [Oli10b, Tro, Tro15]. One important application of matrix concentration is on the spectra of random matrices with independent entries. These random matrices can be represented as matrix series upon a direct entry-wise decomposition, as we show below
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