Related papers: Sharp density discrepancy for cut and project sets…
We study the $L_p$-discrepancy of random point sets in high dimensions, with emphasis on small values of $p$. Although the classical $L_p$-discrepancy suffers from the curse of dimensionality for all $p \in (1,\infty)$, the gap between…
Experimental designs that spread out points apart from each other on projections are important for computer experiments when not necessarily all factors have substantial influence on the response. We provide a theoretical framework to…
In this paper we study the geometric discrepancy of explicit constructions of uniformly distributed points on the two-dimensional unit sphere. We show that the spherical cap discrepancy of random point sets, of spherical digital nets and of…
Counting integer points in large convex bodies with smooth boundaries containing isolated flat points is oftentimes an intermediate case between balls (or convex bodies with smooth boundaries having everywhere positive curvature) and cubes…
The L_2-discrepancy measures the irregularity of the distribution of a finite point set. In this note we prove lower bounds for the L_2 discrepancy of arbitrary N-point sets. Our main focus is on the two-dimensional case. Asymptotic upper…
We consider the Dirichlet problem for elliptic systems with periodically distributed inclusions whose conduction parameter exhibits a significant contrast compared to the background media. We develop a unified method to quantify the…
Dense forests are discrete subsets of Euclidean space which are uniformly close to all sufficiently long line segments. The degree of density of a dense forest is measured by its visibility function. We show that cut-and-project…
In this article, we establish an analogue of the dimension growth conjecture, which is regarding the density of rational points on projective varieties, for compact submanifolds of $\mathbb{R}^n$ with non-vanishing curvature. We also…
We prove explicit bounds on the number of lattice points on or near a convex curve in terms of geometric invariants such as length, curvature, and affine arclength. In several of our results we obtain the best possible constants. Our…
The subgradient projection iteration is a classical method for solving a convex inequality. Motivated by works of Polyak and of Crombez, we present and analyze a more general method for finding a fixed point of a cutter, provided that the…
$f$-divergences are a general class of divergences between probability measures which include as special cases many commonly used divergences in probability, mathematical statistics and information theory such as Kullback-Leibler…
This paper is devoted to the study of a discrepancy-type characteristic -- the fixed volume discrepancy -- of the Fibonacci point set in the unit square. It was observed recently that this new characteristic allows us to obtain optimal rate…
A method for detecting and approximating fault lines or surfaces, respectively, or decision curves in two and three dimensions with guaranteed accuracy is presented. Reformulated as a classification problem, our method starts from a set of…
We prove a general theorem on cutoffs for symmetric simple exclusion processes on graphs with open boundaries, under the natural assumption that the graphs converge geometrically and spectrally to a compact metric measure space with…
Divergence estimators based on direct approximation of density-ratios without going through separate approximation of numerator and denominator densities have been successfully applied to machine learning tasks that involve distribution…
We provide a simple unified approach to obtain (i) Discrete polygonal isoperimetric type inequalities of arbitrary high order. (ii) Arbitrary high order isoperimetric type inequalities for smooth curves, where both upper and lower bounds…
A new lower bound on the average reconstruction error variance of multidimensional sampling and reconstruction is presented. It applies to sampling on arbitrary lattices in arbitrary dimensions, assuming a stochastic process with constant,…
When performing classification tasks, raw high dimensional features often contain redundant information, and lead to increased computational complexity and overfitting. In this paper, we assume the data samples lie on a single underlying…
In this paper, we present a method to determine if a lift-and-project cut for a mixed-integer linear program is irregular, in which case the cut is not equivalent to any intersection cut from the bases of the linear relaxation. This is an…
We undertake a detailed study of the $L^2$ discrepancy of rational and irrational 2-dimensional lattices either with or without symmetrization. We give a full characterization of lattices with optimal $L^2$ discrepancy in terms of the…