N. Tomczak-Jaegermann

activity

19922019

most citedQuantitative estimates of the convergence of the empirical covariance matrix in Log-concave Ensembles

117 citations · 265 across the 28 of their papers we have counts for

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15 papers · 1 filter

math.PR2019

Small ball probability for the condition number of random matrices

Alexander E. Litvak, Konstantin Tikhomirov, Nicole Tomczak-Jaegermann

Let $A$ be an $n \times n$ random matrix with i.i.d. entries of zero mean, unit variance and a bounded subgaussian moment. We show that the condition number $s_{\max}(A)/s_{\min}(A)…

math.PR2018

Circular law for sparse random regular digraphs

Alexander Litvak, Anna Lytova, Konstantin Tikhomirov +2

Fix a constant $C \geq 1$ and let $d = d (n)$ satisfy $d \leq ln^{C} n$ for every large integer $n$ . Denote by $A_{n}$ the adjacency matrix of a uniform random directed $d$ -regular graph…

math.PR2018

The rank of random regular digraphs of constant degree

Alexander Litvak, Anna Lytova, Konstantin Tikhomirov +2

Let $d$ be a fixed large integer. For any $n$ larger than $d$ , let $A_{n}$ be the adjacency matrix of the random directed $d$ -regular graph on $n$ vertices, with the uniform distribu…

math.PR2018

Structure of eigenvectors of random regular digraphs

Alexander Litvak, Anna Lytova, Konstantin Tikhomirov +2

Let $d$ and $n$ be integers satisfying $C \leq d \leq exp (c ln n)$ for some universal constants $c, C > 0$ , and let $z \in C$ . Denote by $M$ the adjacency matrix of a r…

math.PR2017★ 12 cited

The smallest singular value of a shifted $d$ -regular random square matrix

Alexander Litvak, Anna Lytova, Konstantin Tikhomirov +2

We derive a lower bound on the smallest singular value of a random $d$ -regular matrix, that is, the adjacency matrix of a random $d$ -regular directed graph. More precisely, let $C_…

math.PR2015

Adjacency matrices of random digraphs: singularity and anti-concentration

Alexander E. Litvak, Anna Lytova, Konstantin Tikhomirov +2

Let $D_{n, d}$ be the set of all $d$ -regular directed graphs on $n$ vertices. Let $G$ be a graph chosen uniformly at random from $D_{n, d}$ and $M$ be its adjac…

math.PR2015

On the interval of fluctuation of the singular values of random matrices

Olivier Guédon, Alexander E. Litvak, Alain Pajor +1

Let $A$ be a matrix whose columns $X_{1}, \dots, X_{N}$ are independent random vectors in $R^{n}$ . Assume that the tails of the 1-dimensional marginals decay as $\mathbb{P}(|\lan…

math.PR2012★ 15 cited

Moment estimates for convex measures

Radosław Adamczak, Olivier Guédon, Rafał Latała +4

Let $p \geq 1$ , $\eps >0$, $r\geq (1+\eps) p$, and $X$ be a $(- 1/ r)$ -concave random vector in $R^{n}$ with Euclidean norm $∣ X ∣$ . We prove that $(\E |X|^{p})^{1/{p}}\leq c (C(\eps) \E…

math.PR2012★ 25 cited

A short proof of Paouris' inequality

Radosław Adamczak, Rafał Latała, Alexander E. Litvak +3

We give a short proof of a result of G. Paouris on the tail behaviour of the Euclidean norm $∣ X ∣$ of an isotropic log-concave random vector $X \in R^{n}$ , stating that for every $t\ge…

math.PR2011★ 14 cited

Tail estimates for norms of sums of log-concave random vectors

Radosław Adamczak, Rafał Latała, Alexander E. Litvak +2

We establish new tail estimates for order statistics and for the Euclidean norms of projections of an isotropic log-concave random vector. More generally, we prove tail estimates f…

math.PR2011★ 4 cited

Chevet type inequality and norms of submatrices

Radosław Adamczak, Rafał Latała, Alexander E. Litvak +2

We prove a Chevet type inequality which gives an upper bound for the norm of an isotropic log-concave unconditional random matrix in terms of expectation of the supremum of "symmet…

math.PR2011★ 12 cited

Geometry of log-concave Ensembles of random matrices and approximate reconstruction

Radosław Adamczak, Rafał Latała, Alexander E. Litvak +2

We study the Restricted Isometry Property of a random matrix $Γ$ with independent isotropic log-concave rows. To this end, we introduce a parameter $Γ_{k, m}$ that controls uniforml…

math.PR2010

Sharp bounds on the rate of convergence of the empirical covariance matrix

Radosław Adamczak, Alexander E. Litvak, Alain Pajor +1

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 probabilit…

math.PR2009★ 17 cited

Restricted isometry property of matrices with independent columns and neighborly polytopes by random sampling

Radosław Adamczak, Alexander E. Litvak, Alain Pajor +1

This paper considers compressed sensing matrices and neighborliness of a centrally symmetric convex polytope generated by vectors $\pm X_{1}, ..., \pm X_{N} \in R^{n}$ , ( $N \geq n$ ). We intro…

math.PR2009★ 117 cited

Quantitative estimates of the convergence of the empirical covariance matrix in Log-concave Ensembles

Radosław Adamczak, Alexander E. Litvak, Alain Pajor +1

Let $K$ be an isotropic convex body in $R^{n}$ . Given $\eps>0$, how many independent points $X_{i}$ uniformly distributed on $K$ are needed for the empirical covariance matrix to appr…

Small ball probability for the condition number of random matrices

Circular law for sparse random regular digraphs

The rank of random regular digraphs of constant degree

Structure of eigenvectors of random regular digraphs

The smallest singular value of a shifted d-regular random square matrix

Adjacency matrices of random digraphs: singularity and anti-concentration

On the interval of fluctuation of the singular values of random matrices

Moment estimates for convex measures

A short proof of Paouris' inequality

Tail estimates for norms of sums of log-concave random vectors

Chevet type inequality and norms of submatrices

Geometry of log-concave Ensembles of random matrices and approximate reconstruction

Sharp bounds on the rate of convergence of the empirical covariance matrix

Restricted isometry property of matrices with independent columns and neighborly polytopes by random sampling

Quantitative estimates of the convergence of the empirical covariance matrix in Log-concave Ensembles

The smallest singular value of a shifted $d$ -regular random square matrix