Searcharxiv⌕ Search

arXiv subjects

Grigoris Paouris

Publications and source records attributed to Grigoris Paouris.

At least 37 records · Page 2Linked to original sources
arXiv2017

On Dvoretzky's theorem for subspaces of $L_p$

We prove that for any $2 \varepsilon \mathbb E\|Z\| \right) \leq C \exp \left (- c \min \left\{ α_p \varepsilon^2 n, (\varepsilon n)^{2/p} \right\} \right), \quad 0<\varepsilon<1 , \] where $Z$ is a standard $n$-dimensional Gaussian vectors, $α_p>0$ is a constant depending only on $p$ and $C,c>0$ are absolute constants. As a consequence we show optimal lower bound for the dimension of almost spherical sections for these spaces. In particular, for any $2 0$ is a constant depending only on $p$. This improves upon the previously known estimate due to Figiel, Lindenstrauss and V. Milman.

math.FA↗
arXiv2017

Variance estimates and almost Euclidean structure

We introduce and initiate the study of new parameters associated with any norm and any log-concave measure on $\mathbb R^n$, which provide sharp distributional inequalities. In the Gaussian context this investigation sheds light to the importance of the statistical measures of dispersion of the norm in connection with the local structure of the ambient space. As a byproduct of our study, we provide a short proof of Dvoretzky's theorem which not only supports the aforementioned significance but also complements the classical probabilistic formulation.

math.FA↗
arXiv2017

A Gaussian small deviation inequality for convex functions

Let $Z$ be an $n$-dimensional Gaussian vector and let $f: \mathbb R^n \to \mathbb R$ be a convex function. We show that: $$\mathbb P \left( f(Z) \leq \mathbb E f(Z) -t\sqrt{ {\rm Var} f(Z)} \right) \leq \exp(-ct^2),$$ for all $t>1$, where $c>0$ is an absolute constant. As an application we derive variance-sensitive small ball probabilities for Gaussian processes.

math.PR↗
arXiv2017

Probabilistic Condition Number Estimates For Real Polynomial Systems I: A Broader Family Of Distributions

We consider the sensitivity of real roots of polynomial systems with respect to perturbations of the coefficients. In particular - for a version of the condition number defined by Cucker, Krick, Malajovich, and Wschebor - we establish new probabilistic estimates that allow a much broader family of measures than considered earlier. We also generalize further by allowing over-determined systems. Along the way, we derive new Lipshitz estimates for polynomial maps from R^n to R^m, extending earlier work of Kellog on the case m=1, which may be of independent interest. In Part II, we study smoothed complexity and how sparsity (in the sense of restricting which monomial terms can appear) can help further improve earlier condition number estimates.

math.PR↗
arXiv2017

Random version of Dvoretzky's theorem in $\ell_p^n$

We study the dependence on $\varepsilon$ in the critical dimension $k(n,p,\varepsilon)$ for which one can find random sections of the $\ell_p^n$-ball which are $(1+\varepsilon)$-spherical. We give lower (and upper) estimates for $k(n,p,\varepsilon)$ for all eligible values $p$ and $\varepsilon$ as $n\to \infty$, which agree with the sharp estimates for the extreme values $p=1$ and $p=\infty$. Toward this end, we provide tight bounds for the Gaussian concentration of the $\ell_p$-norm.

math.FA↗
arXiv2017

On a quantitative reversal of Alexandrov's inequality

Alexandrov's inequalities imply that for any convex body $A$, the sequence of intrinsic volumes $V_1(A),\ldots,V_n(A)$ is non-increasing (when suitably normalized). Milman's random version of Dvoretzky's theorem shows that a large initial segment of this sequence is essentially constant, up to a critical parameter called the Dvoretzky number. We show that this near-constant behavior actually extends further, up to a different parameter associated with $A$. This yields a new quantitative reverse inequality that sits between the approximate reverse Urysohn inequality, due to Figiel--Tomczak-Jaegermann and Pisier, and the sharp reverse Urysohn inequality for zonoids, due to Hug--Schneider. In fact, we study concentration properties of the volume radius and mean width of random projections of $A$ and show how these lead naturally to such reversals.

math.MG↗
arXiv2016

Randomized isoperimetric inequalities

We discuss isoperimetric inequalities for convex sets. These include the classical isoperimetric inequality and that of Brunn-Minkowski, Blaschke-Santalo, Busemann-Petty and their various extensions. We show that many such inequalities admit stronger randomized forms in the following sense: for natural families of associated random convex sets one has stochastic dominance for various functionals such as volume, surface area, mean width and others. By laws of large numbers, these randomized versions recover the classical inequalities. We give an overview of when such stochastic dominance arises and its applications in convex geometry and probability.

math.MG↗
arXiv2016

Random ball-polyhedra and inequalities for intrinsic volumes

We prove a randomized version of the generalized Urysohn inequality relating mean-width to the other intrinsic volumes. To do this, we introduce a stochastic approximation procedure that sees each convex body K as the limit of intersections of Euclidean balls of large radii and centered at randomly chosen points. The proof depends on a new isoperimetric inequality for the intrinsic volumes of such intersections. If the centers are i.i.d. and sampled according to a bounded continuous distribution, then the extremizing measure is uniform on a Euclidean ball. If one additionally assumes that the centers have i.i.d. coordinates, then the uniform measure on a cube is the extremizer. We also discuss connections to a randomized version of the extended isoperimetric inequality and symmetrization techniques.

math.MG↗
arXiv2016

An interpolation proof of Ehrhard's inequality

We prove Ehrhard's inequality using interpolation along the Ornstein-Uhlenbeck semi-group. We also provide an improved Jensen inequality for Gaussian variables that might be of independent interest.

math.PR↗
arXiv2016

On sharp bounds for marginal densities of product measures

We discuss optimal constants in a recent result of Rudelson and Vershynin on marginal densities. We show that if $f$ is a probability density on $\R^n$ of the form $f(x)=\prod_{i=1}^n f_i(x_i)$, where each $f_i$ is a density on $\R$, say bounded by one, then the density of any marginal $π_E(f)$ is bounded by $2^{k/2}$, where $k$ is the dimension of $E$. The proof relies on an adaptation of Ball's approach to cube slicing, carried out for functions. Motivated by inequalities for dual affine quermassintegrals, we also prove an isoperimetric inequality for certain averages of the marginals of such $f$ for which the cube is the extremal case.

math.PR↗
arXiv2015

Bounding marginal densities via affine isoperimetry

Let $μ$ be a probability measure on $\mathbb{R}^n$ with a bounded density $f$. We prove that the marginals of $f$ on most subspaces are well-bounded. For product measures, studied recently by Rudelson and Vershynin, our results show there is a trade-off between the strength of such bounds and the probability with which they hold. Our proof rests on new affinely-invariant extremal inequalities for certain averages of $f$ on the Grassmannian and affine Grassmannian. These are motivated by Lutwak's dual affine quermassintegrals for convex sets. We show that key invariance properties of the latter, due to Grinberg, extend to families of functions. The inequalities we obtain can be viewed as functional analogues of results due to Busemann--Straus, Grinberg and Schneider. As an application, we show that without any additional assumptions on $μ$, any marginal $π_E(μ)$, or a small perturbation thereof, satisfies a nearly optimal small-ball probability.

math.PR↗
arXiv2013

Volume of the polar of random sets and shadow systems

We obtain optimal inequalities for the volume of the polar of random sets, generated for instance by the convex hull of independent random vectors in Euclidean space. Extremizers are given by random vectors uniformly distributed in Euclidean balls. This provides a random extension of the Blaschke-Santalo inequality which, in turn, can be derived by the law of large numbers. The method involves generalized shadow systems, their connection to Busemann type inequalities, and how they interact with functional rearrangement inequalities.

math.MG↗
arXiv2013

Improved Hölder and reverse Hölder inequalities for correlated Gaussian random vectors

We propose algebraic criteria that yield sharp Hölder types of inequalities for the product of functions of Gaussian random vectors with arbitrary covariance structure. While our lower inequality appears to be new, we prove that the upper inequality gives an equivalent formulation for the geometric Brascamp-Lieb inequality for Gaussian measures. As an application, we retrieve the Gaussian Hypercontractivity as well as its reverse and we present a generalization of the sharp Young and reverse Young inequalities. From the latter, we recover several known inequalities in literatures including the Prékopa-Leindler and Barthe inequalities.

math.PR↗
arXiv2012

A central limit theorem for projections of the cube

We prove a central limit theorem for the volume of projections of the N-cube onto a random subspace of dimension n, when n is fixed and N tends to infinity. Randomness in this case is with respect to the Haar measure on the Grassmannian manifold.

math.PR↗
arXiv2011

On generic chaining and the smallest singular value of random matrices with heavy tails

We present a very general chaining method which allows one to control the supremum of the empirical process $\sup_{h \in H} |N^{-1}\sum_{i=1}^N h^2(X_i)-\E h^2|$ in rather general situations. We use this method to establish two main results. First, a quantitative (non asymptotic) version of the classical Bai-Yin Theorem on the singular values of a random matrix with i.i.d entries that have heavy tails, and second, a sharp estimate on the quadratic empirical process when $H=\{\inr{t,\cdot} : t \in T\}$, $T \subset \R^n$ and $μ$ is an isotropic, unconditional, log-concave measure.

math.PR↗