scholarly journals Sharp lower bounds on the least singular value of a random matrix without the fourth moment condition

2015 ◽  
Vol 20 (0) ◽  
Author(s):  
Pavel Yaskov
Keyword(s):  
Lower Bounds ◽  
Random Matrix ◽  
Singular Value ◽  
Moment Condition ◽  
Fourth Moment ◽  
Least Singular Value

scholarly journals On minimal singular values of random matrices with correlated entries

2015 ◽  
Vol 04 (02) ◽  
pp. 1550006 ◽  
Author(s):  
F. Götze ◽  
A. Naumov ◽  
A. Tikhomirov
Keyword(s):  
Random Matrices ◽  
Random Matrix ◽  
Singular Values ◽  
Singular Value ◽  
The Matrix ◽  
Class Of Matrices ◽  
Least Singular Value ◽  
Mutually Independent

Let X be a random matrix whose pairs of entries Xjk and Xkj are correlated and vectors (Xjk, Xkj), for 1 ≤ j < k ≤ n, are mutually independent. Assume that the diagonal entries are independent from off-diagonal entries as well. We assume that [Formula: see text], for any j, k = 1, …, n and 𝔼 XjkXkj = ρ for 1 ≤ j < k ≤ n. Let Mn be a non-random n × n matrix with ‖Mn‖ ≤ KnQ, for some positive constants K > 0 and Q ≥ 0. Let sn(X + Mn) denote the least singular value of the matrix X + Mn. It is shown that there exist positive constants A and B depending on K, Q, ρ only such that [Formula: see text] As an application of this result we prove the elliptic law for this class of matrices with non-identically distributed correlated entries.

scholarly journals The strong circular law: A combinatorial view

2020 ◽  
pp. 2150031 ◽  
Author(s):  
Vishesh Jain
Keyword(s):  
Error Rate ◽  
Random Matrix ◽  
Random Variable ◽  
Singular Value ◽  
Operator Norm ◽  
Additive Combinatorics ◽  
Counting Problem ◽  
Circular Law ◽  
Matrix Formula ◽  
Least Singular Value

Let [Formula: see text] be an [Formula: see text] complex random matrix, each of whose entries is an independent copy of a centered complex random variable [Formula: see text] with finite nonzero variance [Formula: see text]. The strong circular law, proved by Tao and Vu, states that almost surely, as [Formula: see text], the empirical spectral distribution of [Formula: see text] converges to the uniform distribution on the unit disc in [Formula: see text]. A crucial ingredient in the proof of Tao and Vu, which uses deep ideas from additive combinatorics, is controlling the lower tail of the least singular value of the random matrix [Formula: see text] (where [Formula: see text] is fixed) with failure probability that is inverse polynomial. In this paper, using a simple and novel approach (in particular, not using machinery from additive combinatorics or any net arguments), we show that for any fixed complex matrix [Formula: see text] with operator norm at most [Formula: see text] and for all [Formula: see text], [Formula: see text] where [Formula: see text] is the least singular value of [Formula: see text] and [Formula: see text] are positive absolute constants. Our result is optimal up to the constants [Formula: see text] and the inverse exponential-type error rate improves upon the inverse polynomial error rate due to Tao and Vu. Our proof relies on the solution to the so-called counting problem in inverse Littlewood–Offord theory, developed by Ferber, Luh, Samotij, and the author, a novel complex anti-concentration inequality, and a “rounding trick” based on controlling the [Formula: see text] operator norm of heavy-tailed random matrices.

scholarly journals On singular values distribution of a matrix large auto-covariance in the ultra-dimensional regime

2015 ◽  
Vol 04 (04) ◽  
pp. 1550015
Author(s):  
Qinwen Wang ◽  
Jianfeng Yao
Keyword(s):  
Singular Values ◽  
Singular Value ◽  
Moment Condition ◽  
Fourth Moment ◽  
Limiting Behavior ◽  
The Moment ◽  
Fixed Positive Integer ◽  
The Right ◽  
Auto Covariance ◽  
Random Limit

Let [Formula: see text] be a sequence of independent real random vectors of [Formula: see text]-dimension and let [Formula: see text] be the lag-[Formula: see text] ([Formula: see text] is a fixed positive integer) auto-covariance matrix of [Formula: see text]. This paper investigates the limiting behavior of the singular values of [Formula: see text] under the so-called ultra-dimensional regime where [Formula: see text] and [Formula: see text] in a related way such that [Formula: see text]. First, we show that the singular value distribution of [Formula: see text], after a suitable normalization, converges to a non-random limit [Formula: see text] (quarter law) under the fourth-moment condition. Second, we establish the convergence of its largest singular value to the right edge of the support of [Formula: see text]. Both results are derived using the moment method.

An inverse scattering method in time-domain electromagnetics

1981 ◽  
Vol 59 (10) ◽  
pp. 1348-1353
Author(s):  
Sujeet K. Chaudhuri
Keyword(s):  
Inverse Scattering ◽  
Time Domain ◽  
Random Noise ◽  
Power Series Expansion ◽  
Inverse Scattering Method ◽  
Wave Number ◽  
Moment Condition ◽  
Scattering Method ◽  
Fourth Moment ◽  
Rayleigh Coefficient

An inverse scattering model, based on the time-domain concepts of electromagnetic theory is developed. Using the first five (zeroth to fourth) moment condition integrals, the Rayleigh coefficient and the next higher order nonzero coefficient of the power series expansion in k (wave number) of the object backscattering response are recovered. The Rayleigh coefficient and the other coefficient thus recovered are used (with the ellipsoidal assumption for the object shape) to determine the dimensions and orientation of the object.Some numerical results of the application of this coefficient recovery technique to conducting ellipsoidal scatterers are presented. The performance of the software system in the presence of normally distributed random noise is also studied.

scholarly journals Local Semicircle Law Under Fourth Moment Condition

2019 ◽  
Vol 33 (3) ◽  
pp. 1327-1362
Author(s):  
F. Götze ◽  
A. Naumov ◽  
A. Tikhomirov
Keyword(s):  
Moment Condition ◽  
Fourth Moment ◽  
Semicircle Law ◽  
Local Semicircle Law

On updating the least singular value: a lower bound

CALCOLO ◽  
2003 ◽  
Vol 40 (4) ◽  
pp. 213-229 ◽  
Author(s):  
C. Fassino
Keyword(s):  
Lower Bound ◽  
Singular Value ◽  
Least Singular Value
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