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… Expand
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 generalExpand
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 complexExpand
Constructive Regularization of the Random Matrix Norm
  • E. Rebrova
  • Mathematics
  • Journal of Theoretical Probability
  • 2019
We study the structure of $$n \times n$$ n × n random matrices with centered i.i.d. entries having only two finite moments. In the recent joint work with R. Vershynin, we have shown that the operatorExpand
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. Expand
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 aExpand
Singularity of sparse Bernoulli matrices
Let $M_n$ be an $n\times n$ random matrix with i.i.d. Bernoulli(p) entries. We show that there is a universal constant $C\geq 1$ such that, whenever $p$ and $n$ satisfy $C\log n/n\leq p\leq C^{-1}$,Expand
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 weExpand
Upper bound for intermediate singular values of random matrices
In this paper, we prove that an $n\times n$ matrix $A$ with independent centered subgaussian entries satisfies \[ s_{n+1-l}(A) \le C_1t \frac{l}{\sqrt{n}} \] with probability at leastExpand
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, byExpand
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, andExpand
...
1
2
3
4
...

References

SHOWING 1-10 OF 42 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 randomExpand
Sharp bounds on the rate of convergence of the empirical covariance matrix
Let $X_1,..., X_N\in\R^n$ be independent centered random vectors with log-concave distribution and with the identity as covariance matrix. We show that with overwhelming probability at least $1 - 3Expand
The limit of the smallest singular value of random matrices with i.i.d. entries
Let $\{a_{ij}\}$ $(1\le i,j<\infty)$ be i.i.d. real valued random variables with zero mean and unit variance and let an integer sequence $(N_m)_{m=1}^\infty$ satisfy $m/N_m\longrightarrow z$ for someExpand
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 $\sqrt{n}$. Expand
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 theExpand
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 theExpand
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 areExpand
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 problemExpand
The Littlewood-Offord problem and invertibility of random matrices
Abstract We prove two basic conjectures on the distribution of the smallest singular value of random n × n matrices with independent entries. Under minimal moment assumptions, we show that theExpand
Concentration and regularization of random graphs
TLDR
This paper studies how close random graphs are typically to their expectations through the concentration of the adjacency and Laplacian matrices in the spectral norm, and builds a new decomposition of random graphs based on Grothendieck-Pietsch factorization. Expand
...
1
2
3
4
5
...