Related papers: Cohomological VC-density: Bounds and Applications
Among cocomplete categories, the locally presentable ones can be defined as those with a strong generator consisting of presentable objects. Assuming Vop{\v{e}}nka's Principle, we prove that a cocomplete category is locally presentable iff…
New Vapnik and Chervonenkis type concentration inequalities are derived for the empirical distribution of an independent random sample. Focus is on the maximal deviation over classes of Borel sets within a low probability region. The…
We introduce a new ladder of function spaces which is shown to fill in the gap between the weak $L^{p\infty}$ spaces and the larger Morrey spaces, $M^p$. Our motivation for introducing these new spaces, denoted $\V^{pq}$, is to gain a more…
We derive that dpR(n) \leq dens(n) \leq dpR(n)+1, where dens(n) is the supremum of the VC density of all formulas in n parameters, and dpR(n) is the maximum depth of an ICT pattern in n variables. Consequently, strong dependence is…
We propose a new class of hypertopologies, called here weak$^{\ast }$ hypertopologies, on the dual space $\mathcal{X}^{\ast }$ of a real or complex topological vector space $\mathcal{X}$. The most well-studied and well-known hypertopology…
For any $Q\in\{\frac{3}{2},2,\frac{5}{2},3,\dotsc\}$, we establish a structure theory for the class $\mathcal{S}_Q$ of stable codimension 1 stationary integral varifolds admitting no classical singularities of density $<Q$. This theory…
Despite the fact that copulas are commonly considered as analytically smooth/regular objects, derivatives of copulas have to be handled with care. Triggered by a recently published result characterizing multivariate copulas via…
Let K be a field that is complete with respect to a nonarchimedean absolute value such that K has a countable dense subset. We prove that the Berkovich analytification V^an of any d-dimensional quasi-projective scheme V over K embeds in…
Approximating complex probability densities is a core problem in modern statistics. In this paper, we introduce the concept of Variational Inference (VI), a popular method in machine learning that uses optimization techniques to estimate…
In this short note we give a glimpse of homotopy type theory, a new field of mathematics at the intersection of algebraic topology and mathematical logic, and we explain Vladimir Voevodsky's univalent interpretation of it. This…
Voronoi constellations (VCs) are finite sets of vectors of a coding lattice enclosed by the translated Voronoi region of a shaping lattice, which is a sublattice of the coding lattice. In conventional VCs, the shaping lattice is a scaled-up…
In this work we study the quantitative relation between VC-dimension and two other basic parameters related to learning and teaching. Namely, the quality of sample compression schemes and of teaching sets for classes of low VC-dimension.…
The Sutherland approximation to the van der Waals forces is applied to the derivation of a self-consistent Vlasov-type field in a liquid filling a half space, bordering vacuum. The ensuing Vlasov equation is then derived, and solved to…
The concept of convex compactness, weaker than the classical notion of compactness, is introduced and discussed. It is shown that a large class of convex subsets of topological vector spaces shares this property and that is can be used in…
We are interested in attracting sets of $\mathbb{P}^k(\mathbb{C})$ which are of small topological degree and of codimension $1.$ We first show that there exists a large family of examples. Then we study their ergodic and pluripotential…
We combine the fundamental results of Breuillard, Green, and Tao on the structure of approximate groups, together with "tame" arithmetic regularity methods based on work of the authors and Terry, to give a structure theorem for finite…
We study the generalization capacity of group convolutional neural networks. We identify precise estimates for the VC dimensions of simple sets of group convolutional neural networks. In particular, we find that for infinite groups and…
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…
In these expository notes, intended for students without background in point-set topology, we develop the basic theory of the Stone-Cech compactification without reference to open sets, closed sets, filters, or nets. In particular, this…
It is shown how tools from the area of Model Theory, specifically from the Theory of o-minimality, can be used to prove that a class of functions is VC-subgraph (in the sense of Dudley, 1987), and therefore satisfies a uniform polynomial…