arXiv subjects
Publications and source records attributed to Grigoris Paouris.
We introduce an empirical version of dual volumes within dual Brunn--Minkowski theory. These provide a simple model for computation and analysis under symmetrization, including moment inequalities. As an application, we derive a new proof of the Busemann intersection inequality that relies on counting points in slabs rather than working directly with volumes of slices.
We show that there exists a constant $C > 0$ such that for any $n \geq 1$ and any convex body $K \subset \mathbf{R}^n$, \[ 1 \leq \inf_{T \in \mathrm{GL}(n)} \, \frac{M^\ast(TK)}{\mathrm{vr}(TK)} \leq C\sqrt{\log(\mathrm{e} n)}, \] where $M^\ast$ denotes the spherical mean width and $\mathrm{vr}(\cdot)$ denotes the volume radius. The righthand side is attained, up to universal constants, by the crosspolytope and the regular $n$-simplex. Analogously, we show that, up to universal constants, the logarithm of the Euclidean covering number is maximized over convex bodies $K \subset \mathbf{R}^n$ by the simplex and crosspolytope. Our proof makes use of Eldan's stochastic localization.
The problem of detecting fake data inspires the following seemingly simple mathematical question. Sample a data point $X$ from the standard normal distribution in $\mathbb{R}^n$. An adversary observes $X$ and corrupts it by adding a vector $rt$, where they can choose any vector $t$ from a fixed set $T$ of the adversary's ``tricks'', and where $r>0$ is a fixed radius. The adversary's choice of $t=t(X)$ may depend on the true data $X$. The adversary wants to hide the corruption by making the fake data $X+rt$ statistically indistinguishable from the real data $X$. What is the largest radius $r=r(T)$ for which the adversary can create an undetectable fake? We show that for highly symmetric sets $T$, the detectability radius $r(T)$ is approximately twice the scaled Gaussian width of $T$. The upper bound actually holds for arbitrary sets $T$ and generalizes to arbitrary, non-Gaussian distributions of real data $X$. The lower bound may fail for not highly symmetric $T$, but we conjecture that this problem can be solved by considering the focused version of the Gaussian width of $T$, which focuses on the most important directions of $T$.
The Petty projection inequality is a fundamental affine isoperimetric principle for convex sets. It has shaped several directions of research in convex geometry which forged new connections between projection bodies, centroid bodies, and mixed volume inequalities. We establish several different empirical forms of the Petty projection inequality by re-examining these key relationships from a stochastic perspective. In particular, we derive sharp extremal inequalities for several multiple-entry functionals of random convex sets, including mixed projection bodies and mixed volumes.
We study the small ball probability of an order-$\ell$ simple random tensor $X=X^{(1)}\otimes\cdots\otimes X^{(\ell)}$ where $X^{(i)}, 1\leq i\leq\ell$ are independent random vectors in $\mathbb{R}^n$ that are log-concave or have independent coordinates with bounded densities. We show that the probability that the projection of $X$ onto an $m$-dimensional subspace $F$ falls within an Euclidean ball of length $\varepsilon$ is upper bounded by $\frac{\varepsilon}{(\ell-1)!}\left(C\log\left(\frac{e}{\varepsilon}\right)\right)^{\ell}$ and also this upper bound is sharp when $m$ is small. We also established that a much better estimate holds true for a random subspace.
This note addresses the question of optimally estimating a linear functional of an object acquired through linear observations corrupted by random noise, where optimality pertains to a worst-case setting tied to a symmetric, convex, and closed model set containing the object. It complements the article "Statistical Estimation and Optimal Recovery" published in the Annals of Statistics in 1994. There, Donoho showed (among other things) that, for Gaussian noise, linear maps provide near-optimal estimation schemes relatively to a performance measure relevant in Statistical Estimation. Here, we advocate for a different performance measure arguably more relevant in Optimal Recovery. We show that, relatively to this new measure, linear maps still provide near-optimal estimation schemes even if the noise is merely log-concave. Our arguments, which make a connection to the deterministic noise situation and bypass properties specific to the Gaussian case, offer an alternative to parts of Donoho's proof.
We establish a family of isoperimetric inequalities for sets that interpolate between intersection bodies and dual Lp centroid bodies. This provides a bridge between the Busemann intersection inequality and the Lutwak--Zhang inequality. The approach depends on new empirical versions of these inequalities.
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.
We consider the sensitivity of real zeros of structured polynomial systems to perturbations of their coefficients. In particular, we provide explicit estimates for condition numbers of structured random real polynomial systems, and extend these estimates to the smoothed analysis setting.
We study the complexity of approximating complex zero sets of certain $n$-variate exponential sums. We show that the real part, $R$, of such a zero set can be approximated by the $(n-1)$-dimensional skeleton, $T$, of a polyhedral subdivision of $\mathbb{R}^n$. In particular, we give an explicit upper bound on the Hausdorff distance: $Δ(R,T) =O\left(t^{3.5}/δ\right)$, where $t$ and $δ$ are respectively the number of terms and the minimal spacing of the frequencies of $g$. On the side of computational complexity, we show that even the $n=2$ case of the membership problem for $R$ is undecidable in the Blum-Shub-Smale model over $\mathbb{R}$, whereas membership and distance queries for our polyhedral approximation $T$ can be decided in polynomial-time for any fixed $n$.
This article concerns the non-asymptotic analysis of the singular values (and Lyapunov exponents) of Gaussian matrix products in the regime where $N,$ the number of term in the product, is large and $n,$ the size of the matrices, may be large or small and may depend on $N$. We obtain concentration estimates for sums of Lyapunov exponents, a quantitative rate of convergence of the empirical measure of the normalized squared singular values to the uniform distribution on $[0,1]$, and results on the joint normality of Lyapunov exponents when $N$ is sufficiently large as a function of $n.$ Our technique consists of non-asymptotic versions of the ergodic theory approach at $N=\infty$ due originally to Furstenberg and Kesten in the 1960's, which were then further developed by Newman and Isopi-Newman as well as by a number of other authors in the 1980's. Our key technical idea is that small ball probabilities for volumes of random projections give a way to quantify convergence in the multiplicative ergodic theorem for random matrices.
In this note we study a conjecture of Madiman and Wang which predicted that the generalized Gaussian distribution minimizes the Rényi entropy of the sum of independent random variables. Through a variational analysis, we show that the generalized Gaussian fails to be a minimizer for the problem.
We prove new versions of the isomorphic Busemann-Petty problem for two different measures and show how these results can be used to recover slicing and distance inequalities. We also prove a sharp upper estimate for the outer volume ratio distance from an arbitrary convex body to the unit balls of subspaces of $L_p$.
Recovery procedures in various application in Data Science are based on \emph{stable point separation}. In its simplest form, stable point separation implies that if $f$ is "far away" from $0$, and one is given a random sample $(f(Z_i))_{i=1}^m$ where a proportional number of the sample points may be corrupted by noise, that information is still enough to exhibit that $f$ is far from $0$. Stable point separation is well understood in the context of iid sampling, and to explore it for general sampling methods we introduce a new notion---the \emph{coordinate small-ball} of a random vector $X$. Roughly put, this feature captures the number of "relatively large coordinates" of $(| |)_{i=1}^m$, where $T:\mathbb{R}^n \to \mathbb{R}^m$ is an arbitrary linear operator and $(u_i)_{i=1}^m$ is any fixed orthonormal basis of $\mathbb{R}^m$. We show that under the bare-minimum assumptions on $X$, and with high probability, many of the values $| |$ are at least of the order $\|T\|_{S_2}/\sqrt{m}$. As a result, the "coordinate structure" of $TX$ exhibits the typical Euclidean norm of $TX$ and does so in a stable way. One outcome of our analysis is that random sub-sampled convolutions satisfy stable point separation under minimal assumptions on the generating random vector---a fact that was known previously only in a highly restrictive setup, namely, for random vectors with iid subgaussian coordinates.
Building on work of Furstenberg and Tzkoni, we introduce ${\bf r}$-flag affine quermassintegrals and their dual versions. These quantities generalize affine and dual affine quermassintegrals as averages on flag manifolds (where the Grassmannian can be considered as a special case). We establish affine and linear invariance properties and extend fundamental results to this new setting. In particular, we prove several affine isoperimetric inequalities from convex geometry and their approximate reverse forms. We also introduce functional forms of these quantities and establish corresponding inequalities.
Suppose $F:=(f_1,\ldots,f_n)$ is a system of random $n$-variate polynomials with $f_i$ having degree $\leq\!d_i$ and the coefficient of $x^{a_1}_1\cdots x^{a_n}_n$ in $f_i$ being an independent complex Gaussian of mean $0$ and variance $\frac{d_i!}{a_1!\cdots a_n!\left(d_i-\sum^n_{j=1}a_j \right)!}$. Recent progress on Smale's 17th Problem by Lairez --- building upon seminal work of Shub, Beltran, Pardo, Bürgisser, and Cucker --- has resulted in a deterministic algorithm that finds a single (complex) approximate root of $F$ using just $N^{O(1)}$ arithmetic operations on average, where $N\!:=\!\sum^n_{i=1}\frac{(n+d_i)!}{n!d_i!}$ ($=n(n+\max_i d_i)^{O(\min\{n,\max_i d_i)\}}$) is the maximum possible total number of monomial terms for such an $F$. However, can one go faster when the number of terms is smaller, and we restrict to real coefficient and real roots? And can one still maintain average-case polynomial-time with more general probability measures? We show the answer is yes when $F$ is instead a binomial system --- a case whose numerical solution is a key step in polyhedral homotopy algorithms for solving arbitrary polynomial systems. We give a deterministic algorithm that finds a real approximate root (or correctly decides there are none) using just $O(n^2(\log(n)+\log\max_i d_i))$ arithmetic operations on average. Furthermore, our approach allows Gaussians with arbitrary variance. We also discuss briefly the obstructions to maintaining average-case time polynomial in $n\log \max_i d_i$ when $F$ has more terms.
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.
We propose a program for establishing a conjectural extension to the class of (origin-symmetric) log-concave probability measures $μ$, of the classical dual Sudakov Minoration on the expectation of the supremum of a Gaussian process: \begin{equation} \label{eq:abstract} M(Z_p(μ), C \int ||x||_K dμ\cdot K) \leq \exp(C p) \;\;\, \forall p \geq 1 . \end{equation} Here $K$ is an origin-symmetric convex body, $Z_p(μ)$ is the $L_p$-centroid body associated to $μ$, $M(A,B)$ is the packing-number of $B$ in $A$, and $C > 0$ is a universal constant. The Program consists of first establishing a Weak Generalized Dual Sudakov Minoration, involving the dimension $n$ of the ambient space, which is then self-improved to a dimension-free estimate after applying a dimension-reduction step. The latter step may be thought of as a conjectural "small-ball one-sided" variant of the Johnson--Lindenstrauss dimension-reduction lemma. We establish the Weak Generalized Dual Sudakov Minoration for a variety of log-concave probability measures and convex bodies (for instance, this step is fully resolved assuming a positive answer to the Slicing Problem). The Separation Dimension-Reduction step is fully established for ellipsoids and, up to logarithmic factors in the dimension, for cubes, resulting in a corresponding Generalized (regular) Dual Sudakov Minoration estimate for these bodies and arbitrary log-concave measures, which are shown to be (essentially) best-possible. Along the way, we establish a regular version of (\ref{eq:abstract}) for all $p \geq n$ and provide a new direct proof of Sudakov Minoration via The Program.