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Petros Valettas

Publications and source records attributed to Petros Valettas.

15 recordsLinked to original sources
arXiv2025

A probabilistic approach to strong natural boundaries

We study the local non-extendability of random power series beyond their disk of convergence. We show that random power series formed by independent coefficients which are asymptotically anti-concentrated admit the circle of radius of convergence as strong natural boundary, even in a Nevanlinna sense. Our results extend previous work of Breuer and Simon (2011) for the case of independent coefficients. Our motivation stems from the study of Padé approximants of random power series as a denoising tool.

math.PR↗
arXiv2025

Decompositions of finite high-dimensional random arrays

A $d$-dimensional random array on a nonempty set $I$ is a stochastic process $\boldsymbol{X}=\langle X_s:s\in \binom{I}{d}\rangle$ indexed by the set $\binom{I}{d}$ of all $d$-element subsets of $I$. We obtain structural decompositions of finite, high-dimensional random arrays whose distribution is invariant under certain symmetries. Our first main result is a distributional decomposition of finite, (approximately) spreadable, high-dimensional random arrays whose entries take values in a finite set; the two-dimensional case of this result is the finite version of an infinitary decomposition due to Fremlin and Talagrand. Our second main result is a physical decomposition of finite, spreadable, high-dimensional random arrays with square-integrable entries that is the analogue of the Hoeffding/Efron--Stein decomposition. All proofs are effective. We also present applications of these decompositions in the study of concentration of functions of finite, high-dimensional random arrays.

math.PR↗
arXiv2024

On the clustering of Padé zeros and poles of random power series

We estimate non-asymptotically the probability of uniform clustering around the unit circle of the zeros of the $[m,n]$-Padé approximant of a random power series $f(z) = \sum_{j=0}^\infty a_j z^j$ for $a_j$ independent, with finite first moment, and Lévy function satisfying $L(a_j , \varepsilon) \leq K\varepsilon$. Under the same assumptions we show that almost surely $f$ has infinitely many zeros in the unit disc, with the unit circle serving as a natural boundary for $f$. For $R_m$ the radius of the largest disc containing at most $m$ zeros of $f$, a deterministic result of Edrei implies that in our setting the poles of the $[m,n]$-Padé approximant almost surely cluster uniformly at the circle of radius $R_m$ as $n \to \infty$ and $m$ stays fixed, and we provide almost sure rates of converge of these $R_m$'s to $1$. We also show that our results on the clustering of the zeros hold for log-concave vectors $(a_j)$ with not necessarily independent coordinates.

math.CV↗
arXiv2023

Concentration estimates for functions of finite high-dimensional random arrays

Let $\boldsymbol{X}$ be a $d$-dimensional random array on $[n]$ whose entries take values in a finite set $\mathcal{X}$, that is, $\boldsymbol{X}=\langle X_s:s\in \binom{[n]}{d}\rangle$ is an $\mathcal{X}$-valued stochastic process indexed by the set $\binom{[n]}{d}$ of all $d$-element subsets of $[n]:=\{1,\dots,n\}$. We give easily checked conditions on $\boldsymbol{X}$ that ensure, for instance, that for every function $f\colon \mathcal{X}^{\binom{[n]}{d}}\to\mathbb{R}$ that satisfies $\mathbb{E}[f(\boldsymbol{X})]=0$ and $\|f(\boldsymbol{X})\|_{L_p}=1$ for some $p>1$, the random variable $f(\boldsymbol{X})$ becomes concentrated after conditioning it on a large subarray of $\boldsymbol{X}$. These conditions cover several classes of random arrays with not necessarily independent entries. Applications are given in combinatorics, and examples are also presented that show the optimality of various aspects of the results.

math.PR↗
arXiv2023

Approximate Real Symmetric Tensor Rank

We investigate the effect of an $\varepsilon$-room of perturbation tolerance on symmetric tensor decomposition. To be more precise, suppose a real symmetric $d$-tensor $f$, a norm $||.||$ on the space of symmetric $d$-tensors, and $\varepsilon >0$ are given. What is the smallest symmetric tensor rank in the $\varepsilon$-neighborhood of $f$? In other words, what is the symmetric tensor rank of $f$ after a clever $\varepsilon$-perturbation? We prove two theorems and develop three corresponding algorithms that give constructive upper bounds for this question. With expository goals in mind; we present probabilistic and convex geometric ideas behind our results, reproduce some known results, and point out open problems.

math.NA↗
arXiv2021

Hypercontractivity and Lower Deviation Estimates in Normed Spaces

We consider the problem of estimating small ball probabilities $\mathbb P\{f(G) \leqslant δ\mathbb Ef(G)\}$ for sub-additive,positively homogeneous functions $f$ with respect to the Gaussian measure. We establish estimates that depend on global parameters of the underlying function which take into account analytic and statistical measures, such as the variance and the $L^1$-norms of its partial derivatives. This leads to dimension-dependent bounds for small ball and lower small deviation estimates for seminorms when the linear structure is appropriately chosen to optimize the aforementioned parameters. Our bounds are best possible up to numerical constants. In all regimes, $\|G\|_\infty = \max_{ i \leqslant n}|g_i|$ arises as an extremal case in this study. The proofs exploit the convexity and hypercontractivity properties of the Gaussian measure.

math.FA↗
arXiv2018

Dichotomies, structure, and concentration in normed spaces

We use probabilistic, topological and combinatorial methods to establish the following deviation inequality: For any normed space $X=(\mathbb R^n ,\|\cdot\| )$ there exists an invertible linear map $T:\mathbb R^n \to \mathbb R^n$ with \[ \mathbb P\left( \big| \|TG\| -\mathbb E\|TG\| \big| > \varepsilon \mathbb E\|TG\| \right) \leq C\exp \left( -c\max\{ \varepsilon^2, \varepsilon \} \log n \right),\quad \varepsilon>0, \] where $G$ is the standard $n$-dimensional Gaussian vector and $C,c>0$ are universal constants. It follows that for every $\varepsilon\in (0,1)$ and for every normed space $X=(\mathbb R^n,\|\cdot\|)$ there exists a $k$-dimensional subspace of $X$ which is $(1+\varepsilon)$-Euclidean and $k\geq c\varepsilon \log n/\log\frac{1}{\varepsilon}$. This improves by a logarithmic on $\varepsilon$ term the best previously known result due to G. Schechtman.

math.FA↗
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

On the tightness of Gaussian concentration for convex functions

The concentration of measure phenomenon in Gauss' space states that every $L$-Lipschitz map $f$ on $\mathbb R^n$ satisfies \[ γ_{n} \left(\{ x : | f(x) - M_{f} | \geqslant t \} \right) \leqslant 2 e^{ - \frac{t^2}{ 2L^2} }, \quad t>0, \] where $γ_{n} $ is the standard Gaussian measure on $\mathbb R^{n}$ and $M_{f}$ is a median of $f$. In this work, we provide necessary and sufficient conditions for when this inequality can be reversed, up to universal constants, in the case when $f$ is additionally assumed to be convex. In particular, we show that if the variance ${\rm Var}(f)$ (with respect to $γ_{n}$) satisfies $ αL \leqslant \sqrt{ {\rm Var}(f) } $ for some $ 0<α\leqslant 1$, then \[ γ_{n} \left(\{ x : | f(x) - M_{f} | \geqslant t \}\right) \geqslant c e^{ -C \frac{t^2}{ L^2} } , \quad t>0 ,\] where $c,C>0$ are constants depending only on $α$.

math.PR↗
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

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

Inequalities for the surface area of projections of convex bodies

We provide general inequalities that compare the surface area S(K) of a convex body K in ${\mathbb R}^n$ to the minimal, average or maximal surface area of its hyperplane or lower dimensional projections. We discuss the same questions for all the quermassintegrals of K. We examine separately the dependence of the constants on the dimension in the case where K is in some of the classical positions or K is a projection body. Our results are in the spirit of the hyperplane problem, with sections replaced by projections and volume by surface area.

math.MG↗