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Paweł Wolff

Bio: Paweł Wolff is an academic researcher from Polish Academy of Sciences. The author has contributed to research in topics: Sobolev inequality & Lipschitz continuity. The author has an hindex of 8, co-authored 18 publications receiving 341 citations. Previous affiliations of Paweł Wolff include University of Zaragoza & University of Warsaw.

Papers
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Journal ArticleDOI
TL;DR: In this paper, a concentration inequality for not necessarily Lipschitz functions with bounded derivatives of higher orders was proposed, which holds when the underlying measure satisfies a family of Sobolev type inequalities.
Abstract: Building on the inequalities for homogeneous tetrahedral polynomials in independent Gaussian variables due to R. Latala we provide a concentration inequality for not necessarily Lipschitz functions $$f:\mathbb {R}^n \rightarrow \mathbb {R}$$ with bounded derivatives of higher orders, which holds when the underlying measure satisfies a family of Sobolev type inequalities $$\begin{aligned} \Vert g- \mathbb Eg\Vert _p \le C(p)\Vert abla g\Vert _p. \end{aligned}$$ Such Sobolev type inequalities hold, e.g., if the underlying measure satisfies the log-Sobolev inequality (in which case $$C(p) \le C\sqrt{p}$$ ) or the Poincare inequality (then $$C(p) \le Cp$$ ). Our concentration estimates are expressed in terms of tensor-product norms of the derivatives of $$f$$ . When the underlying measure is Gaussian and $$f$$ is a polynomial (not necessarily tetrahedral or homogeneous), our estimates can be reversed (up to a constant depending only on the degree of the polynomial). We also show that for polynomial functions, analogous estimates hold for arbitrary random vectors with independent sub-Gaussian coordinates. We apply our inequalities to general additive functionals of random vectors (in particular linear eigenvalue statistics of random matrices) and the problem of counting cycles of fixed length in Erdős–Renyi random graphs, obtaining new estimates, optimal in a certain range of parameters.

82 citations

Posted Content
TL;DR: In this article, a concentration inequality for non-necessarily Lipschitz functions with bounded derivatives of higher orders was proposed, where the underlying measure satisfies a family of Sobolev type inequalities.
Abstract: Building on the inequalities for homogeneous tetrahedral polynomials in independent Gaussian variables due to R. Latala we provide a concentration inequality for non-necessarily Lipschitz functions $f\colon \R^n \to \R$ with bounded derivatives of higher orders, which hold when the underlying measure satisfies a family of Sobolev type inequalities $\|g- \E g\|_p \le C(p)\| abla g\|_p.$ Such Sobolev type inequalities hold, e.g., if the underlying measure satisfies the log-Sobolev inequality (in which case $C(p) \le C\sqrt{p}$) or the Poincare inequality (then $C(p) \le Cp$). Our concentration estimates are expressed in terms of tensor-product norms of the derivatives of $f$. When the underlying measure is Gaussian and $f$ is a polynomial (non-necessarily tetrahedral or homogeneous), our estimates can be reversed (up to a constant depending only on the degree of the polynomial). We also show that for polynomial functions, analogous estimates hold for arbitrary random vectors with independent sub-Gaussian coordinates. We apply our inequalities to general additive functionals of random vectors (in particular linear eigenvalue statistics of random matrices) and the problem of counting cycles of fixed length in Erdős-R{e}nyi random graphs, obtaining new estimates, optimal in a certain range of parameters.

61 citations

Journal ArticleDOI
TL;DR: An exchangeable random matrix is a random matrix with distribution invariant under any permutation of the entries, and it is shown, as the dimension tends to infinity, that the empirical spectral distribution tends to the uniform law on the unit disc.
Abstract: An exchangeable random matrix is a random matrix with distribution invariant under any permutation of the entries. For such random matrices, we show, as the dimension tends to infinity, that the empirical spectral distribution tends to the uniform law on the unit disc. This is an instance of the universality phenomenon known as the circular law, for a model of random matrices with dependent entries, rows, and columns. It is also a non-Hermitian counterpart of a result of Chatterjee on the semi-circular law for random Hermitian matrices with exchangeable entries. The proof relies in particular on a reduction to a simpler model given by a random shuffle of a rigid deterministic matrix, on Hermitization, and also on combinatorial concentration of measure and combinatorial Central Limit Theorem. A crucial step is a polynomial bound on the smallest singular value of exchangeable random matrices, which may be of independent interest.

48 citations

Journal ArticleDOI
TL;DR: In this article, the spectral gap and logarithmic Sobolev constants of canonical Gibbs measures associated to gamma distributions with parameter α ≥ 1 were derived for simplices and unit balls of l p n spaces.

36 citations


Cited by
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Book
05 Jun 2014
TL;DR: This text gives a thorough overview of Boolean functions, beginning with the most basic definitions and proceeding to advanced topics such as hypercontractivity and isoperimetry, and includes a "highlight application" such as Arrow's theorem from economics.
Abstract: Boolean functions are perhaps the most basic objects of study in theoretical computer science. They also arise in other areas of mathematics, including combinatorics, statistical physics, and mathematical social choice. The field of analysis of Boolean functions seeks to understand them via their Fourier transform and other analytic methods. This text gives a thorough overview of the field, beginning with the most basic definitions and proceeding to advanced topics such as hypercontractivity and isoperimetry. Each chapter includes a "highlight application" such as Arrow's theorem from economics, the Goldreich-Levin algorithm from cryptography/learning theory, Hstad's NP-hardness of approximation results, and "sharp threshold" theorems for random graph properties. The book includes roughly 450 exercises and can be used as the basis of a one-semester graduate course. It should appeal to advanced undergraduates, graduate students, and researchers in computer science theory and related mathematical fields.

867 citations

Journal ArticleDOI
TL;DR: In this paper, a modern proof of Hanson-Wright inequality for quadratic forms in sub-gaussian random variables is given, and a useful concentration inequality for sub-Gaussian random vectors is given.
Abstract: In this expository note, we give a modern proof of Hanson-Wright inequality for quadratic forms in sub-gaussian random variables.We deduce a useful concentration inequality for sub-gaussian random vectors.Two examples are given to illustrate these results: a concentration of distances between random vectors and subspaces, and a bound on the norms of products of random and deterministic matrices.

687 citations

MonographDOI
01 Jan 2014

575 citations

MonographDOI
01 Jan 2016
TL;DR: All those interested in discrete mathematics, computer science or applied probability and their applications will find this an ideal introduction to the subject.
Abstract: From social networks such as Facebook, the World Wide Web and the Internet, to the complex interactions between proteins in the cells of our bodies, we constantly face the challenge of understanding the structure and development of networks. The theory of random graphs provides a framework for this understanding, and in this book the authors give a gentle introduction to the basic tools for understanding and applying the theory. Part I includes sufficient material, including exercises, for a one semester course at the advanced undergraduate or beginning graduate level. The reader is then well prepared for the more advanced topics in Parts II and III. A final part provides a quick introduction to the background material needed. All those interested in discrete mathematics, computer science or applied probability and their applications will find this an ideal introduction to the subject.

565 citations