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Rafał Latała

Publications and source records attributed to Rafał Latała.

At least 19 recordsLinked to original sources
arXiv2026

Operator $\ell_p\to\ell_q$ norms of Gaussian matrices

We confirm the conjecture posed by Guédon, Hinrichs, Litvak, and Prochno in 2017 that $\mathbb{E}\|(a_{ij}g_{ij})_{i\le m, j\le n}\colon \ell_p^n \to \ell_q^m\|$ is comparable, up to constants depending only on $p$ and $q$, to \[ \max_i \|(a_{ij})_j\|_{p^*} +\max_j \|(a_{ij})_i\|_{q} +\mathbb{E} \max_{i,j} |a_{ij}g_{ij}| \] provided that $1\le p \le 2\le q \le \infty$. This was known before only in the case $p=1$ or $q=\infty$, and in the spectral case $p=2=q$. We also reprove the conjecture in the case $p=2=q$ without using spectral theory (which was employed in the previously known proof).

math.PR↗
arXiv2025

On the spectral norm of Rademacher matrices

We discuss two-sided non-asymptotic bounds for the mean spectral norm of nonhomogenous weighted Rademacher matrices. We show that the recently formulated conjecture holds up to $\log \log \log n$ factor for arbitrary $n\times n$ Rademacher matrices and the triple logarithm may be eliminated for matrices with $\{0,1\}$-coefficients.

math.PR↗
arXiv2024

Operator $\ell_p\to\ell_q$ norms of random matrices with iid entries

We prove that for every $p,q\in[1,\infty]$ and every random matrix $X=(X_{i,j})_{i\le m, j\le n}$ with iid centered entries satisfying the regularity assumption $\|X_{i,j}\|_{2ρ} \le α\|X_{i,j}\|_ρ$ for every $ρ\ge 1$, the expectation of the operator norm of $X$ from $\ell_p^n$ to $\ell_q^m$ is comparable, up to a constant depending only on $α$, to \[ m^{1/q}\sup_{t\in B_p^n}\Bigl\|\sum_{j=1}^nt_jX_{1,j}\Bigr\|_{ q\wedge \operatorname{Log} m} +n^{1/p^*}\sup_{s\in B_{q^*}^m}\Bigl\|\sum_{i=1}^{m} s_iX_{i,1}\Bigr\|_{ p^*\wedge \operatorname{Log} n}. \] We give more explicit formulas, expressed as exact functions of $p$, $q$, $m$, and $n$, for the asymptotic operator norms in the case when the entries $X_{i,j}$ are: Gaussian, Weibullian, log-concave tailed, and log-convex tailed. In the range $1\le q\le 2\le p$ we provide two-sided bounds under a weaker regularity assumption $(\mathbb{E} X_{1,1}^4)^{1/4}\leq α(\mathbb{E} X_{1,1}^2)^{1/2}$.

math.PR↗
arXiv2023

Chevet-type inequalities for subexponential Weibull variables and estimates for norms of random matrices

We prove two-sided Chevet-type inequalities for independent symmetric Weibull random variables with shape parameter $r\in[1,2]$. We apply them to provide two-sided estimates for operator norms from $\ell_p^n$ to $\ell_q^m$ of random matrices $(a_ib_jX_{i,j})_{i\le m, j\le n}$, in the case when $X_{i,j}$'s are iid symmetric Weibull variables with shape parameter $r\in[1,2]$ or when $X$ is an isotropic log-concave unconditional random matrix. We also show how these Chevet-type inequalities imply two-sided bounds for maximal norms from $\ell_p^n$ to $\ell_q^m$ of submatrices of $X$ in both Weibull and log-concave settings.

math.PR↗
arXiv2022

Norms of Randomized Circulant Matrices

We investigate two-sided bounds for operator norms of random matrices with unhomogenous independent entries. We formulate a lower bound for Rademacher matrices and conjecture that it may be reversed up to a universal constant. We show that our conjecture holds up to $\log\log n$ factor for randomized $n\times n$ circulant matrices and double logarithm may be eliminated under some mild additional assumptions on the coefficients.

math.PR↗
arXiv2022

Bounding suprema of canonical processes via convex hull

We discuss the method of bounding suprema of canonical processes based on the inclusion of their index set into a convex hull of a well-controlled set of points. While the upper bound is immediate, the reverse estimate was established to date only for a narrow class of regular stochastic processes. We show that for specific index sets, including arbitrary ellipsoids, regularity assumptions may be substantially weakened.

math.PR↗
arXiv2020

Bounds on moments of weighted sums of finite Riesz products

Let $n_j$ be a lacunary sequence of integers, such that $n_{j+1}/n_j\geq r$. We are interested in linear combinations of the sequence of finite Riesz products $\prod_{j=1}^N(1+\cos(n_j t))$. We prove that, whenever the Riesz products are normalized in $L^p$ norm ($p\geq 1$) and when $r$ is large enough, the $L^p$ norm of such a linear combination is equivalent to the $\ell^p$ norm of the sequence of coefficients. In other words, one can describe many ways of embedding $\ell^p$ into $L^p$ based on Fourier coefficients. This generalizes to vector valued $L^p$ spaces.

math.FA↗
arXiv2020

Hanson-Wright inequality in Banach spaces

We discuss two-sided bounds for moments and tails of quadratic forms in Gaussian random variables with values in Banach spaces. We state a natural conjecture and show that it holds up to additional logarithmic factors. Moreover in a certain class of Banach spaces (including $L_r$-spaces) these logarithmic factors may be eliminated. As a corollary we derive upper bounds for tails and moments of quadratic forms in subgaussian random variables, which extend the Hanson-Wright inequality.

math.PR↗
arXiv2020

Moments of Gaussian chaoses in Banach spaces

We derive moment and tail estimates for Gaussian chaoses of arbitrary order with values in Banach spaces. We formulate a conjecture regarding two-sided estimates and show that it holds in a certain class of Banach spaces including L_q spaces. As a corollary we obtain two-sided bounds for moments of chaoses with values in L_q spaces based on exponential random variables.

math.PR↗
arXiv2019

Hadamard products and moments of random vectors

We derive new comparison inequalities between weak and strong moments of norms of random vectors with optimal (up to an universal factor) constants. We discuss applications to the concentration of log-concave random vectors and bounds on $p$-summing norms of finite rank operators.

math.PR↗
arXiv2019

Two-sided estimates for order statistics of log-concave random vectors

We establish two-sided bounds for expectations of order statistics ($k$-th maxima) of moduli of coordinates of centered log-concave random vectors with uncorrelated coordinates. Our bounds are exact up to multiplicative universal constants in the unconditional case for all $k$ and in the isotropic case for $k \leq n-cn^{5/6}$. We also derive two-sided estimates for expectations of sums of $k$ largest moduli of coordinates for some classes of random vectors.

math.PR↗
arXiv2018

The dimension-free structure of nonhomogeneous random matrices

Let $X$ be a symmetric random matrix with independent but non-identically distributed centered Gaussian entries. We show that $$ \mathbf{E}\|X\|_{S_p} \asymp \mathbf{E}\Bigg[ \Bigg(\sum_i\Bigg(\sum_j X_{ij}^2\Bigg)^{p/2}\Bigg)^{1/p} \Bigg] $$ for any $2\le p\le\infty$, where $S_p$ denotes the $p$-Schatten class and the constants are universal. The right-hand side admits an explicit expression in terms of the variances of the matrix entries. This settles, in the case $p=\infty$, a conjecture of the first author, and provides a complete characterization of the class of infinite matrices with independent Gaussian entries that define bounded operators on $\ell_2$. Along the way, we obtain optimal dimension-free bounds on the moments $(\mathbf{E}\|X\|_{S_p}^p)^{1/p}$ that are of independent interest. We develop further extensions to non-symmetric matrices and to nonasymptotic moment and norm estimates for matrices with non-Gaussian entries that arise, for example, in the study of random graphs and in applied mathematics.

math.PR↗
arXiv2017

On ${\cal Z}_p$-norms of random vectors

To any $n$-dimensional random vector $X$ we may associate its $L_p$-centroid body ${\cal Z}_p(X)$ and the corresponding norm. We formulate a conjecture concerning the bound on the ${\cal Z}_p(X)$-norm of $X$ and show that it holds under some additional symmetry assumptions. We also relate our conjecture with estimates of covering numbers and Sudakov-type minorization bounds.

math.PR↗
arXiv2016

Comparison of weak and strong moments for vectors with independent coordinates

We show that for $p\ge 1$, the $p$-th moment of suprema of linear combinations of independent centered random variables are comparable with the sum of the first moment and the weak $p$-th moment provided that $2q$-th and $q$-th integral moments of these variables are comparable for all $ q \ge 2$. The latest condition turns out to be necessary in the i.i.d. case.

math.PR↗
arXiv2015

Royen's proof of the Gaussian correlation inequality

We present in detail Thomas Royen's proof of the Gaussian correlation inequality which states that $μ(K\cap L)\geq μ(K)μ(L)$ for any centered Gaussian measure $μ$ on $R^d$ and symmetric convex sets $K,L$ in $R^d$.

math.PR↗
arXiv2015

Weak and strong moments of l_r-norms of log-concave vectors

We show that for $p\geq 1$ and $r\geq 1$ the $p$-th moment of the $l_r$-norm of a log-concave random vector is comparable to the sum of the first moment and the weak $p$-th moment up to a constant proportional to $r$. This extends the previous result of Paouris concerning Euclidean norms.

math.PR↗
arXiv2015

Asymptotic entropic uncertainty relations

We analyze entropic uncertainty relations for two orthogonal measurements on a $N$-dimensional Hilbert space, performed in two generic bases. It is assumed that the unitary matrix $U$ relating both bases is distributed according to the Haar measure on the unitary group. We provide lower bounds on the average Shannon entropy of probability distributions related to both measurements. The bounds are stronger than these obtained with use of the entropic uncertainty relation by Maassen and Uffink, and they are optimal up to additive constants. We also analyze the case of a large number of measurements and obtain strong entropic uncertainty relations which hold with high probability with respect to the random choice of bases. The lower bounds we obtain are optimal up to additive constants and allow us to establish the conjecture by Wehner and Winter on the asymptotic behavior of constants in entropic uncertainty relations as the dimension tends to infinity. As a tool we develop estimates on the maximum operator norm of a submatrix of a fixed size of a random unitary matrix distributed according to the Haar measure, which are of an independent interest.

quant-ph↗