Coverings of random ellipsoids, and invertibility of matrices with i.i.d. heavy-tailed entries

@article{Rebrova2018CoveringsOR,
  title={Coverings of random ellipsoids, and invertibility of matrices with i.i.d. heavy-tailed entries},
  author={Elizaveta Rebrova and Konstantin E. Tikhomirov},
  journal={Israel Journal of Mathematics},
  year={2018},
  volume={227},
  pages={507-544}
}
Let A = (aij) be an n × n random matrix with i.i.d. entries such that Ea11 = 0 and Ea112 = 1. We prove that for any δ > 0 there is L > 0 depending only on δ, and a subset N of B2n of cardinality at most exp(δn) such that with probability very close to one we have $$A\left( {B_2^n} \right)\subset\mathop \cup \limits_{y \in A\left( \mathcal{N} \right)} \left( {y + L\sqrt n B_2^n} \right)$$A(B2n)∪y∈A(N)(y+LnB2n). In fact, a stronger statement holds true. As an application, we show that for some L… 
Approximate Spielman-Teng theorems for random matrices with heavy-tailed entries: a combinatorial view
This paper makes two contributions to the areas of anti-concentration and non-asymptotic random matrix theory. First, we study the counting problem in inverse Littlewood-Offord theory for general
Lower bounds for the smallest singular value of structured random matrices
We obtain lower tail estimates for the smallest singular value of random matrices with independent but non-identically distributed entries. Specifically, we consider $n\times n$ matrices with complex
Constructive Regularization of the Random Matrix Norm
  • E. Rebrova
  • Mathematics, Computer Science
    Journal of Theoretical Probability
  • 2019
TLDR
It is enough to zero out a small fraction of the rows and columns of A with largest $$L_2$$ L 2 norms to bring the operator norm of A to the almost optimal order O ( n log log n ) , under additional assumption that the matrix entries are symmetrically distributed.
Invertibility via distance for noncentered random matrices with continuous distributions
TLDR
The method is principally different from a standard approach involving a decomposition of the unit sphere and coverings, as well as an approach of Sankar-Spielman-Teng for non-centered Gaussian matrices.
Structure of eigenvectors of random regular digraphs
Let $n$ be a large integer, let $d$ satisfy $C\leq d\leq \exp(c\sqrt{\ln n})$ for some universal constants $c, C>0$, and let $z\in {\mathcal C}$. Further, denote by $M$ the adjacency matrix of a
Singularity of sparse Bernoulli matrices
TLDR
There is a universal constant C\geq 1 such that, whenever $p and $n$ satisfy C\log n/n/n\leq p-1, there is a singular value of $M_n$ such that it contains a zero row or column.
Sharp transition of the invertibility of the adjacency matrices of sparse random graphs
We consider three different models of sparse random graphs:~undirected and directed Erdős-Renyi graphs, and random bipartite graph with an equal number of left and right vertices. For such graphs we
Upper bound for intermediate singular values of random matrices
A note on the universality of ESDs of inhomogeneous random matrices
In this short note, we extend the celebrated results of Tao and Vu, and Krishnapur on the universality of empirical spectral distributions to a wide class of inhomogeneous complex random matrices, by
The circular law for sparse non-Hermitian matrices
For a class of sparse random matrices of the form $A_n =(\xi_{i,j}\delta_{i,j})_{i,j=1}^n$, where $\{\xi_{i,j}\}$ are i.i.d.~centered sub-Gaussian random variables of unit variance, and
...
1
2
3
4
...

References

SHOWING 1-10 OF 34 REFERENCES
The smallest singular value of random rectangular matrices with no moment assumptions on entries
Let δ > 1 and β > 0 be some real numbers. We prove that there are positive u, v, N0 depending only on β and δ with the following property: for any N,n such that N ≥ max(N0, δn), any N × n random
On the interval of fluctuation of the singular values of random matrices
TLDR
It is proved that with high probability A/A has the Restricted Isometry Property (RIP) provided that Euclidean norms $|X_i|$ are concentrated around $\sqrt{n}$.
On minimal singular values of random matrices with correlated entries
Let $\mathbf X$ be a random matrix whose pairs of entries $X_{jk}$ and $X_{kj}$ are correlated and vectors $ (X_{jk},X_{kj})$, for $1\le j 0$ and $Q\ge 0$. Let $s_n(\mathbf X+\mathbf M_n)$ denote the
RANDOM MATRICES: THE CIRCULAR LAW
Let x be a complex random variable with mean zero and bounded variance σ2. Let Nn be a random matrix of order n with entries being i.i.d. copies of x. Let λ1, …, λn be the eigenvalues of . Define the
Bounding the smallest singular value of a random matrix without concentration
Given $X$ a random vector in ${\mathbb{R}}^n$, set $X_1,...,X_N$ to be independent copies of $X$ and let $\Gamma=\frac{1}{\sqrt{N}}\sum_{i=1}^N e_i$ be the matrix whose rows are
Inverse Littlewood-Offord theorems and the condition number of random discrete matrices
Consider a random sum r)\V\ + • • • + r]nvn, where 771, . . . , rin are independently and identically distributed (i.i.d.) random signs and vi, . . . , vn are integers. The Littlewood-Offord problem
Random Matrices: the Distribution of the Smallest Singular Values
Let ξ be a real-valued random variable of mean zero and variance 1. Let Mn(ξ) denote the n × n random matrix whose entries are iid copies of ξ and σn(Mn(ξ)) denote the least singular value of Mn(ξ).
...
1
2
3
4
...