Related papers: VC-Dimension and Distance Chains in $\mathbb{F}_q^…

Given a set $X$ and a function $h:X\longrightarrow\{0,1\}$ which labels each element of $X$ with either $0$ or $1$, we may define a function $h^{(s)}$ to measure the similarity of pairs of points in $X$ according to $h$. Specifically, for…

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…

In response to a 1997 problem of M. Vidyasagar, we state a necessary and sufficient condition for distribution-free PAC learnability of a concept class $\mathscr C$ under the family of all non-atomic (diffuse) measures on the domain…

We introduce and study the spherical dimension, a natural topological relaxation of the VC dimension that unifies several results in learning theory where topology plays a key role in the proofs. The spherical dimension is defined by…

The Vapnik-Chervonenkis dimension (in short, VC-dimension) of a graph is defined as the VC-dimension of the set system induced by the neighborhoods of its vertices. We show that every $n$-vertex graph with bounded VC-dimension contains a…

We study computable probably approximately correct (CPAC) learning, where learners are required to be computable functions. It had been previously observed that the Fundamental Theorem of Statistical Learning, which characterizes PAC…

We say that two partial orders on $[n]$ are compatible if there exists a partial order that refines both of them. This compatibility relation induces a natural set system structure between the collection $\mathcal{F}$ of all partial orders…

In many applications of relational learning, the available data can be seen as a sample from a larger relational structure (e.g. we may be given a small fragment from some social network). In this paper we are particularly concerned with…

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…

Given a set $E \subset {\Bbb F}_q^3$, where ${\Bbb F}_q$ is the field with $q$ elements. Consider a set of "classifiers" ${\mathcal H}^3_t(E)=\{h_y: y \in E\}$, where $h_y(x)=1$ if $x \cdot y=t$, $x \in E$, and $0$ otherwise. We are going…

The Vapnik-Chervonenkis (VC) dimension of the set of half-spaces of R^d with frontiers parallel to the axes is computed exactly. It is shown that it is much smaller than the intuitive value of d. A good approximation based on the Stirling's…

Recently, the authors introduced the theory of high-arity PAC learning, which is well-suited for learning graphs, hypergraphs and relational structures. In the same initial work, the authors proved a high-arity analogue of the Fundamental…

The concept of Vapnik-Chervonenkis (VC) density is pivotal across various mathematical fields, including discrete geometry, probability theory and model theory. In this paper, we introduce a topological generalization of VC-density. Let $Y$…

Since its introduction by Vapnik and Chervonenkis in the 1960s, the VC dimension and its variants have played a central role in numerous fields. In this paper, we investigate several variants of the VC dimension and their applications to…

Degrading performance of indexing schemes for exact similarity search in high dimensions has long since been linked to histograms of distributions of distances and other 1-Lipschitz functions getting concentrated. We discuss this…

In response to a 1997 problem of M. Vidyasagar, we state a criterion for PAC learnability of a concept class $\mathscr C$ under the family of all non-atomic (diffuse) measures on the domain $\Omega$. The uniform Glivenko--Cantelli property…

We study the complexity of the Hitting Set problem in set systems (hypergraphs) that avoid certain sub-structures. In particular, we characterize the classical and parameterized complexity of the problem when the Vapnik-Chervonenkis…

For any set system $H=(V,R), \ R \subseteq 2^V$, a subset $S \subseteq V$ is called \emph{shattered} if every $S' \subseteq S$ results from the intersection of $S$ with some set in $\R$. The \emph{VC-dimension} of $H$ is the size of a…

VC-dimension and $\varepsilon$-nets are key concepts in Statistical Learning Theory. Intuitively, VC-dimension is a measure of the size of a class of sets. The famous $\varepsilon$-net theorem, a fundamental result in Discrete Geometry,…

We study the Vapnik-Chervonenkis (VC) dimension of the set of quadratic residues (i.e. squares) in finite fields, $\mathbb F_q$, when considered as a subset of the additive group. We conjecture that as $q \to \infty$, the squares have the…