Tusnády's problem and online two-sample thinning

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1. Très proche Même question Pas pertinent

   Resolving Tusn\'ady's Problem

   Nicola Wengiel · 2026 · Pas encore résumé

   Resolving Tusndy's Problem établit l'ordre optimal de la discrépance des boîtes ancrées, également déterminé ici pour les boîtes parallèles aux axes.

   Résumé scientifique

   We prove a lower bound of $\Omega_d((\log N)^{d-1/2})$ for the worst-case combinatorial discrepancy of $N$-point sets with respect to anchored boxes in every fixed dimension $d\ge3$. Together with Nikolov's upper bound of the same order and Wei's planar lower bound, this settles Tusn\'ady's problem: the discrepancy of axis-parallel boxes is $\Theta_d((\log N)^{d-1/2})$ in every fixed dimension $d\ge2$. The proof combines smoothed product test functions, sampling discretization, and an entropy argument for conditioned Gaussian vectors.

2. Très proche Base de cet article Cité par cet article Utilisé en profondeur par cet article Pas pertinent

   Thinning to Improve Two‐Sample Discrepancy

   Gleb Smirnov, Roman Vershynin · 2025 · Random Structures & Algorithms · cité 4 fois · Pas encore résumé

   Thinning to Improve Two-Sample Discrepancy propose un algorithme en ligne qui réduit la discrépance de deux échantillons en supprimant des points, problème approfondi ici.

   Résumé scientifique

   The discrepancy between two independent samples X1,…,Xn$$ {X}_1,\dots, {X}_n $$ and Y1,…,Yn$$ {Y}_1,\dots, {Y}_n $$ drawn from the same distribution on ℝd$$ {\mathbb{R}}^d $$ typically has order O(n)$$ O\left(\sqrt{n}\right) $$ even in one dimension. We give a simple online algorithm that reduces the discrepancy to O(log2dn)$$ O\left({\log}^{2d}n\right) $$ by discarding a small fraction of the points.

3. Très proche Même méthode, autre problème Cité par cet article Pas pertinent

   The power of online thinning in reducing discrepancy

   Raaz Dwivedi, O. Feldheim, O. Gurel-Gurevich et al. · 2016 · Probability theory and related fields · cité 28 fois · Pas encore résumé

   The power of online thinning in reducing discrepancy étudie l'amincissement en ligne de points uniformes pour contrôler la discrépance, avec une règle de conservation des points.

   Résumé scientifique

   Consider an infinite sequence of independent, uniformly chosen points from [0,1]d\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[0,1]^d$$\end{document}. After looking at each point in the sequence, an overseer is allowed to either keep it or reject it, and this choice may depend on the locations of all previously kept points. However, the overseer must keep at least one of every two consecutive points. We call a sequence generated in this fashion a two-thinning sequence. Here, the purpose of the overseer is to c…

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   Optimal Online Discrepancy Minimization

   Janardhan Kulkarni, Victor Reis, T. Rothvoss · 2023 · Symposium on the Theory of Computing · cité 31 fois · Pas encore résumé

   Optimal Online Discrepancy Minimization équilibre en ligne des sommes préfixes de vecteurs, tandis que cet article contrôle la discrépance préfixe par amincissement de deux échantillons.

   Résumé scientifique

   We prove that there exists an online algorithm that for any sequence of vectors v1,…,vT ∈ ℝn with ||vi||2 ≤ 1, arriving one at a time, decides random signs x1,…,xT ∈ { −1,1} so that for every t ≤ T, the prefix sum ∑i=1t xivi is 10-subgaussian. This improves over the work of Alweiss, Liu and Sawhney who kept prefix sums O(√log(nT))-subgaussian, and gives a O(√logT) bound on the discrepancy maxt ∈ T ||∑i=1t xi vi||∞. Our proof combines a generalization of Banaszczyk’s prefix balancing result to trees with a cloning argument to find distributions rather than single colorings. We also show a matching Ω(√logT) strategy for an oblivious adversary.

5. Proche Base de cet article Cité par cet article Utilisé en profondeur par cet article Pas pertinent

   Tight Bounds for Tusn\'ady's Problem in the Plane

   Zhe-Wei Wei · 2026 · cité 2 fois · Pas encore résumé

   Tight Bounds for Tusndy's Problem in the Plane prouve la borne inférieure en dimension deux que l'article étend à toute dimension fixe.

   Résumé scientifique

   We show that the worst-case combinatorial discrepancy of $n$ points in the plane with respect to axis-parallel rectangles is $\Theta(\log^{3/2}n)$. The known bounds were $\Omega(\log n)$ and $O(\log^{3/2}n)$; we prove the matching lower bound. It holds for random point sets: for every $A>0$, there is a constant $c_A>0$ such that, with probability at least $1-e^{-An}$, every coloring of $n$ independent uniform points in the unit square has an anchored rectangle with imbalance at least $c_A(\log_2n)^{3/2}$. The proof is surprisingly simple and elementary. It reveals one coordinate digit by digit. With overwhelming probability over the points, the conditional gains of an oscillation potential a…

6. Proche Contexte plus large Cité par cet article Pas pertinent

   Tusnády's problem, the transference principle, and non-uniform QMC sampling

   C. Aistleitner, D. Bilyk, Aleksandar Nikolov · 2016 · arXiv.org · preprint, non relu par les pairs · cité 27 fois · Pas encore résumé

   Tusndy's problem, the transference principle, and non-uniform QMC sampling traite de discrépance de points pour des mesures générales, plutôt que d'amincissement en ligne.

   Résumé scientifique

   It is well-known that for every \(N \ge 1\) and \(d \ge 1\) there exist point sets \(x_1, \dots , x_N \in [0,1]^d\) whose discrepancy with respect to the Lebesgue measure is of order at most \((\log N)^{d-1} N^{-1}\). In a more general setting, the first author proved together with Josef Dick that for any normalized measure \(\mu \) on \([0,1]^d\) there exist points \(x_1, \dots , x_N\) whose discrepancy with respect to \(\mu \) is of order at most \((\log N)^{(3d+1)/2} N^{-1}\). The proof used methods from combinatorial mathematics, and in particular a result of Banaszczyk on balancings of vectors. In the present note we use a version of the so-called transference principle together with re…

7. Proche Contexte plus large Pas pertinent

   Discrepancy theory, Tverberg's theorem, and regression depth

   A. Lopez, Pablo Soberón · 2026 · Pas encore résumé

   Discrepancy theory, Tverberg's theorem, and regression depth relie la discrépance à des partitions robustes et à la profondeur de régression, non aux boîtes ni à l'amincissement.

   Résumé scientifique

   We prove new bounds for Tverberg's theorem with tolerance. We show that $N = rt+\Theta_{d,r}(t^{1/2-1/(2d)})$, where $N$ is the smallest number such that any set of $N$ points in $\mathbb{R}^d$ has a partition into $r$ parts such that the convex hulls of the parts intersect even if we remove any $t$ of the points. We extend Tverberg's theorem with tolerance to families of hyperplanes in $\mathbb{R}^d$, and show that for any set of $rt + O_{d,r}(t^{1/2-1/(2d)}\sqrt{\log (t+1)})$ hyperplanes in $\mathbb{R}^d$ there exists a partition of them into $r$ parts such that the regression hulls of the parts intersect even if any $t$ hyperplanes are removed. Our bounds follow from establishing a connec…

8. Proche Contexte plus large Cité par cet article Pas pertinent

   Tighter Bounds for the Discrepancy of Boxes and Polytopes

   Aleksandar Nikolov · 2017 · arXiv.org · preprint, non relu par les pairs · cité 32 fois · Pas encore résumé

   Tighter Bounds for the Discrepancy of Boxes and Polytopes étudie la minimisation combinatoire de discrépance pour des boîtes et des polytopes.

   Résumé scientifique

   Combinatorial discrepancy is a complexity measure of a collection of sets which quantifies how well the sets in the collection can be simultaneously balanced. More precisely, we are given an n-point set $P$, and a collection $\mathcal{F} = \{F_1, ..., F_m\}$ of subsets of $P$, and our goal is color $P$ with two colors, red and blue, so that the largest absolute difference between the number of red elements and the number of blue elements (i.e. the discrepancy) in any $F_i$ is minimized. Combinatorial discrepancy has many applications in mathematics and computer science, including constructions of uniformly distributed point sets, and lower bounds for data structures and private data analysis…

Les travaux antérieurs sur lesquels il s’appuie

1. Apparenté* Base de cet article Utilisé en profondeur par cet article Pas pertinent

   Surveys in Combinatorics, 1989: On the method of bounded differences

   C. McDiarmid · 1989 · cité 2 247 fois · Pas encore résumé

   Surveys in Combinatorics, 1989: On the method of bounded differences est cité comme fondement méthodologique, mais son résumé ne précise pas davantage son objet.

2. Apparenté Contexte plus large Pas pertinent

   Discrepancy minimization via a self-balancing walk

   Ryan Alweiss, Yang P. Liu, Mehtaab Sawhney · 2020 · Symposium on the Theory of Computing · cité 87 fois · Pas encore résumé

   Discrepancy minimization via a self-balancing walk analyse l'équilibrage de vecteurs en ligne par un processus aléatoire, sans comparer deux échantillons.

   Résumé scientifique

   We study discrepancy minimization for vectors in ℝn under various settings. The main result is the analysis of a new simple random process in high dimensions through a comparison argument. As corollaries, we obtain bounds which are tight up to logarithmic factors for online vector balancing against oblivious adversaries, resolving several questions posed by Bansal, Jiang, Singla, and Sinha (STOC 2020), as well as a linear time algorithm for logarithmic bounds for the Komlós conjecture.

3. Apparenté Contexte plus large Utilisé en profondeur par cet article Pas pertinent

   Discrepancy and Fisher information

   Gleb Smirnov, Roman Vershynin · 2026 · cité 6 fois · Pas encore résumé

   Discrepancy and Fisher information garde une marche aléatoire dans un corps convexe en supprimant des pas, plutôt que des observations de deux échantillons.

   Résumé scientifique

   We give an online algorithm that keeps a symmetric random walk inside a convex body by discarding some of its steps. The expected number of discarded steps is controlled by a Fisher-information-type quantity associated with the body. For the cube, this gives a dimension-free bound: a walk with unit Euclidean steps can be kept bounded in all coordinates while discarding only a small constant fraction of the steps on average.

4. Apparenté Même question Pas pertinent

   Regularizing Random Points by Deleting a Few

   D. Bilyk, Stefan Steinerberger · 2025 · Journal of Geometric Analysis · cité 4 fois · Pas encore résumé

   Regularizing Random Points by Deleting a Few étudie la réduction de la discrépance de points uniformes unidimensionnels par suppression d'une partie des points.

   Résumé scientifique

   It is well understood that if one is given a set X⊂[0,1]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X \subset [0,1]$$\end{document} of n independent uniformly distributed random variables, then sup0≤x≤1#X∩[0,x]#X-x≲lognnwithhighprobability.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sup _{0 \le x \le 1} \left|…

5. Apparenté Base de cet article Utilisé en profondeur par cet article Pas pertinent

   A Gaussian small deviation inequality for convex functions

   G. Paouris, P. Valettas · 2016 · arXiv: Probability · preprint, non relu par les pairs · cité 42 fois · Pas encore résumé

   A Gaussian small deviation inequality for convex functions établit des bornes de petites déviations gaussiennes, outil utilisé ici pour les bornes inférieures.

   Résumé scientifique

   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.