Related papers: Optimal Bounds on the VC-dimension

We study range spaces, where the ground set consists of either polygonal curves in $\mathbb{R}^d$ or polygonal regions in the plane that may contain holes and the ranges are balls defined by an elastic distance measure, such as the…

This thesis is a study of large sets of unit vectors in $\cx^n$ such that the absolute value of their standard inner products takes on only a small number of values. We begin with bounds: what is the maximal size of a set of lines with only…

A set-system $S\subseteq \{0,1\}^n$ is cube-ideal if its convex hull can be described by capacity and generalized set covering inequalities. In this paper, we use combinatorics, convex geometry, and polyhedral theory to give exponential…

We give a new proof of VC bounds where we avoid the use of symmetrization and use a shadow sample of arbitrary size. We also improve on the variance term. This results in better constants, as shown on numerical examples. Moreover our bounds…

The problem of constrained clustering has attracted significant attention in the past decades. In this paper, we study the balanced $k$-center, $k$-median, and $k$-means clustering problems where the size of each cluster is constrained by…

This paper presents some finite combinatorics of set systems with applications to model theory, particularly the study of dependent theories. There are two main results. First, we give a way of producing lower bounds on VC_ind-density, and…

We introduce a new type of $n$-dimensional generalization of symmetric $(v,k,\lambda)$ block designs. We prove upper bounds on the dimension $n$ in terms of $v$ and $k$. We also define the corresponding concept of $n$-dimensional difference…

Finding tight bounds on the optimal solution is a critical element of practical solution methods for discrete optimization problems. In the last decade, decision diagrams (DDs) have brought a new perspective on obtaining upper and lower…

We investigate the size of the distance set determined by two subsets of finite dimensional vector spaces over finite fields. A lower bound of the size is given explicitly in terms of cardinalities of the two subsets. As a result, we…

Convex geometries form a subclass of closure systems with unique criticals, or $UC$-systems. We show that the $F$-basis introduced in [1] for $UC$-systems, becomes optimum in convex geometries, in two essential parts of the basis: right…

Computation of confidence sets is central to data science and machine learning, serving as the workhorse of A/B testing and underpinning the operation and analysis of reinforcement learning algorithms. This paper studies the geometry of the…

The geometric dimension $g$ of a Vector Addition System with States (VASS) is the dimension of the vector space generated by cycles in the VASS; this parameter refines the standard dimension $d$, the number of counters. Recently, it was…

The Natarajan dimension is a fundamental tool for characterizing multi-class PAC learnability, generalizing the Vapnik-Chervonenkis (VC) dimension from binary to multi-class classification problems. This work establishes upper bounds on…

Totally symmetric sets are a recently introduced tool for studying homomorphisms between groups. In this paper, we give full classifications of totally symmetric sets in certain families of groups and bound their sizes in others. As a…

Statistical learning theory chiefly studies restricted hypothesis classes, particularly those with finite Vapnik-Chervonenkis (VC) dimension. The fundamental quantity of interest is the sample complexity: the number of samples required to…

One of the founding paradigms of machine learning is that a small number of variables is often sufficient to describe high-dimensional data. The minimum number of variables required is called the intrinsic dimension (ID) of the data.…

We introduce a new combinatorial structure: the superselector. We show that superselectors subsume several important combinatorial structures used in the past few years to solve problems in group testing, compressed sensing, multi-channel…

Let $X$ be a finite set. A family $P$ of subsets of $X$ is called a convex geometry with ground set $X$ if (1) $\emptyset, X\in P$; (2) $A\cap B\in P$ whenever $A,B\in P$; and (3) if $A\in P$ and $A\neq X$, there is an element $\alpha\in…

In many machine learning tasks, a common approach for dealing with large-scale data is to build a small summary, {\em e.g.,} coreset, that can efficiently represent the original input. However, real-world datasets usually contain outliers…

In this paper we present a new bound obtained with the probabilistic method for the solution of the Set Covering problem with unit costs. The bound is valid for problems of fixed dimension, thus extending previous similar asymptotic…