arXiv subjects

Nicole Tomczak-Jaegermann

Publications and source records attributed to Nicole Tomczak-Jaegermann.

At least 19 recordsLinked to original sources

arXiv2019

Small ball probability for the condition number of random matrices

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 $s_{\max}(A)/s_{\min}(A)$ satisfies the small ball probability estimate $${\mathbb P}\big\{s_{\max}(A)/s_{\min}(A)\leq n/t\big\}\leq 2\exp(-c t^2),\quad t\geq 1,$$ where $c>0$ may only depend on the subgaussian moment. Although the estimate can be obtained as a combination of known results and techniques, it was not noticed in the literature before. As a key step of the proof, we apply estimates for the singular values of $A$, ${\mathbb P}\big\{s_{n-k+1}(A)\leq ck/\sqrt{n}\big\}\leq 2 \exp(-c k^2), \quad 1\leq k\leq n,$ obtained (under some additional assumptions) by Nguyen.

math.PR↗

arXiv2018

Structure of eigenvectors of random regular digraphs

Let $d$ and $n$ be integers satisfying $C\leq d\leq \exp(c\sqrt{\ln n})$ for some universal constants $c, C>0$, and let $z\in \mathbb{C}$. Denote by $M$ the adjacency matrix of a random $d$-regular directed graph on $n$ vertices. In this paper, we study the structure of the kernel of submatrices of $M-z\,{\rm Id}$, formed by removing a subset of rows. We show that with large probability the kernel consists of two non-intersecting types of vectors, which we call very steep and gradual with many levels. As a corollary, we show, in particular, that every eigenvector of $M$, except for constant multiples of $(1,1,\dots,1)$, possesses a weak delocalization property: its level sets have cardinality less than $Cn\ln^2 d/\ln n$. For a large constant $d$ this provides a principally new structural information on eigenvectors, implying that the number of their level sets grows to infinity with $n$. As a key technical ingredient of our proofs we introduce a decomposition of $\mathbb{C}^n$ into vectors of different degrees of `structuredness', which is an alternative to the decomposition based on the least common denominator in the regime when the underlying random matrix is very sparse.

math.PR↗

arXiv2018

The rank of random regular digraphs of constant degree

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 distribution. We show that $A_n$ has rank at least $n-1$ with probability going to one as $n$ goes to infinity. The proof combines the method of simple switchings and a recent result of the authors on delocalization of eigenvectors of $A_n$.

math.PR↗

arXiv2018

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

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 $C_1<d< c_1 n/\log^2 n$ and let $\mathcal{M}_{n,d}$ be the set of all $0/1$-valued square $n\times n$ matrices such that each row and each column of a matrix $M\in \mathcal{M}_{n,d}$ has exactly $d$ ones. Let $M$ be uniformly distributed on $\mathcal{M}_{n,d}$. Then the smallest singular value $s_{n} (M)$ of $M$ is greater than $c_2 n^{-6}$ with probability at least $1-C_2\log^2 d/\sqrt{d}$, where $c_1$, $c_2$, $C_1$, and $C_2$ are absolute positive constants independent of any other parameters.

math.PR↗

arXiv2018

Circular law for sparse random regular digraphs

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 on $n$ vertices. We show that, as long as $d\to\infty$ with $n$, the empirical spectral distribution of appropriately rescaled matrix $A_n$ converges weakly in probability to the circular law. This result, together with an earlier work of Cook, completely settles the problem of weak convergence of the empirical distribution in directed $d$-regular setting with the degree tending to infinity. As a crucial element of our proof, we develop a technique of bounding intermediate singular values of $A_n$ based on studying random normals to rowspaces and on constructing a product structure to deal with the lack of independence between the matrix entries.

math.PR↗

arXiv2016

Adjacency matrices of random digraphs: singularity and anti-concentration

Let ${\mathcal D}_{n,d}$ be the set of all $d$-regular directed graphs on $n$ vertices. Let $G$ be a graph chosen uniformly at random from ${\mathcal D}_{n,d}$ and $M$ be its adjacency matrix. We show that $M$ is invertible with probability at least $1-C\ln^{3} d/\sqrt{d}$ for $C\leq d\leq cn/\ln^2 n$, where $c, C$ are positive absolute constants. To this end, we establish a few properties of $d$-regular directed graphs. One of them, a Littlewood-Offord type anti-concentration property, is of independent interest. Let $J$ be a subset of vertices of $G$ with $|J|\approx n/d$. Let $δ_i$ be the indicator of the event that the vertex $i$ is connected to $J$ and define $δ= (δ_1, δ_2, ..., δ_n)\in \{0, 1\}^n$. Then for every $v\in\{0,1\}^n$ the probability that $δ=v$ is exponentially small. This property holds even if a part of the graph is "frozen".

math.PR↗

arXiv2016

On the geometry of projective tensor products

In this work, we study the volume ratio of the projective tensor products $\ell^n_p\otimes_π\ell_q^n\otimes_π\ell_r^n$ with $1\leq p\leq q \leq r \leq \infty$. We obtain asymptotic formulas that are sharp in almost all cases. As a consequence of our estimates, these spaces allow for a nearly Euclidean decomposition of Kashin type whenever $1\leq p \leq q\leq r \leq 2$ or $1\leq p \leq 2 \leq r \leq \infty$ and $q=2$. Also, from the Bourgain-Milman bound on the volume ratio of Banach spaces in terms of their cotype $2$ constant, we obtain information on the cotype of these $3$-fold projective tensor products. Our results naturally generalize to $k$-fold products $\ell_{p_1}^n\otimes_π\dots \otimes_π\ell_{p_k}^n$ with $k\in\mathbb N$ and $1\leq p_1 \leq \dots\leq p_k \leq \infty$.

math.FA↗

arXiv2015

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

Let $A$ be a matrix whose columns $X_1,\dots, X_N$ are independent random vectors in $\mathbb{R}^n$. Assume that the tails of the 1-dimensional marginals decay as $\mathbb{P}(|\langle X_i, a\rangle|\geq t)\leq t^{-p}$ uniformly in $a\in S^{n-1}$ and $i\leq N$. Then for $p>4$ we prove that with high probability $A/{\sqrt{n}}$ has the Restricted Isometry Property (RIP) provided that Euclidean norms $|X_i|$ are concentrated around $\sqrt{n}$. We also show that the covariance matrix is well approximated by the empirical covariance matrix and establish corresponding quantitative estimates on the rate of convergence in terms of the ratio $n/N$. Moreover, we obtain sharp bounds for both problems when the decay is of the type $ \exp({-t^α})$ with $α\in (0,2]$, extending the known case $α\in[1, 2]$.

math.PR↗

arXiv2012

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 $1 - 3 \exp(-c\sqrt{n}\r)$ one has $ \sup_{x\in S^{n-1}} \Big|\frac{1/N}\sum_{i=1}^N (| |^2 - \E| |^2\r)\Big| \leq C \sqrt{\frac{n/N}},$ where $C$ is an absolute positive constant. This result is valid in a more general framework when the linear forms $( )_{i\leq N, x\in S^{n-1}}$ and the Euclidean norms $(|X_i|/\sqrt n)_{i\leq N}$ exhibit uniformly a sub-exponential decay. As a consequence, if $A$ denotes the random matrix with columns $(X_i)$, then with overwhelming probability, the extremal singular values $λ_{\rm min}$ and $λ_{\rm max}$ of $AA^\top$ satisfy the inequalities $ 1 - C\sqrt{n/N} \le {λ_{\rm min}/N} \le \frac{λ_{\rm max}/N} \le 1 + C\sqrt{n/N} $ which is a quantitative version of Bai-Yin theorem \cite{BY} known for random matrices with i.i.d. entries.

math.PR↗

arXiv2012

On approximations by projections of polytopes with few facets

We provide an affirmative answer to a problem posed by Barvinok and Veomett, showing that in general an n-dimensional convex body cannot be approximated by a projection of a section of a simplex of a sub-exponential dimension. Moreover, we establish a lower bound of the Banach-Mazur distance between n-dimensional projections of sections of an N-dimensional simplex and a certain convex symmetric body, which is sharp up to a logarithmic factor for all N>n.

math.FA↗

arXiv2012

Moment estimates for convex measures

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 $(\E |X|^{p})^{1/{p}}\leq c (C(\eps) \E|X|+σ_{p}(X))$, where $σ_{p}(X)=\sup_{|z|\leq 1}(\E| |^{p})^{1/p}$, $C(\eps)$ depends only on $\eps$ and $c$ is a universal constant. Moreover, if in addition $X$ is centered then $(\E |X|^{-p})^{-1/{p}}\geq c(\eps) (\E|X| - C σ_{p}(X))$.

math.PR↗

arXiv2012

A short proof of Paouris' inequality

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 $t\geq 1$, $P(|X|\geq ct\sqrt n)\leq \exp(-t\sqrt n)$. More precisely we show that for any log-concave random vector $X$ and any $p\geq 1$, $(E|X|^p)^{1/p}\sim E |X|+\sup_{z\in S^{n-1}}(E |< z,X>|^p)^{1/p}$.

math.PR↗

arXiv2011

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

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 for the norms of projections of sums of independent log-concave random vectors, and uniform versions of these in the form of tail estimates for operator norms of matrices and their sub-matrices in the setting of a log-concave ensemble. This is used to study a quantity $A_{k,m}$ that controls uniformly the operator norm of the sub-matrices with $k$ rows and $m$ columns of a matrix $A$ with independent isotropic log-concave random rows. We apply our tail estimates of $A_{k,m}$ to the study of Restricted Isometry Property that plays a major role in the Compressive Sensing theory.

math.PR↗

arXiv2011

Chevet type inequality and norms of submatrices

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 "symmetric exponential" processes compared to the Gaussian ones in the Chevet inequality. This is used to give sharp upper estimate for a quantity $Γ_{k,m}$ that controls uniformly the Euclidean operator norm of the sub-matrices with $k$ rows and $m$ columns of an isotropic log-concave unconditional random matrix. We apply these estimates to give a sharp bound for the Restricted Isometry Constant of a random matrix with independent log-concave unconditional rows. We show also that our Chevet type inequality does not extend to general isotropic log-concave random matrices.

math.PR↗

arXiv2011

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

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 uniformly the operator norm of sub-matrices with $k$ rows and $m$ columns. This parameter is estimated by means of new tail estimates of order statistics and deviation inequalities for norms of projections of an isotropic log-concave vector.

math.PR↗

arXiv2009

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

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 approximate the identity up to $\eps$ with overwhelming probability? Our paper answers this question posed by Kannan, Lovasz and Simonovits. More precisely, let $X\in\R^n$ be a centered random vector with a log-concave distribution and with the identity as covariance matrix. An example of such a vector $X$ is a random point in an isotropic convex body. We show that for any $\eps>0$, there exists $C(\eps)>0$, such that if $N\sim C(\eps) n$ and $(X_i)_{i\le N}$ are i.i.d. copies of $X$, then $ \Big\|\frac{1}{N}\sum_{i=1}^N X_i\otimes X_i - \Id\Big\| \le ε, $ with probability larger than $1-\exp(-c\sqrt n)$.

math.PR↗

arXiv2009

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

This paper considers compressed sensing matrices and neighborliness of a centrally symmetric convex polytope generated by vectors $\pm X_1,...,\pm X_N\in\R^n$, ($N\ge n$). We introduce a class of random sampling matrices and show that they satisfy a restricted isometry property (RIP) with overwhelming probability. In particular, we prove that matrices with i.i.d. centered and variance 1 entries that satisfy uniformly a sub-exponential tail inequality possess this property RIP with overwhelming probability. We show that such "sensing" matrices are valid for the exact reconstruction process of $m$-sparse vectors via $\ell_1$ minimization with $m\le Cn/\log^2 (cN/n)$. The class of sampling matrices we study includes the case of matrices with columns that are independent isotropic vectors with log-concave densities. We deduce that if $K\subset \R^n$ is a convex body and $X_1,..., X_N\in K$ are i.i.d. random vectors uniformly distributed on $K$, then, with overwhelming probability, the symmetric convex hull of these points is an $m$-centrally-neighborly polytope with $m\sim n/\log^2 (cN/n)$.

math.PR↗

arXiv2008

On the nontrivial projection problem

The Nontrivial Projection Problem asks whether every finite-dimensional normed space of dimension greater than one admits a well-bounded projection of non-trivial rank and corank or, equivalently, whether every centrally symmetric convex body (of arbitrary dimension greater than one) is approximately affinely equivalent to a direct product of two bodies of non-trivial dimension. We show that this is true "up to a logarithmic factor."

math.FA↗