Related papers: Exact VC-Dimensions of Certain Geometric Set Syste…
We announce two breakthrough results concerning important questions in the Theory of Computational Complexity. In this expository paper, a systematic and comprehensive geometric characterization of the Subset Sum Problem is presented. We…
The symmetry dimension of a geometric structure is the dimension of its symmetry algebra. We investigate symmetries of almost quaternionic structures of quaternionic dimension $n$. The maximal possible symmetry is realized by the…
We show that the sets in a family with finite VC dimension can be uniformly approximated within a given error by a finite partition. Immediate corollaries include the fact that VC classes have finite bracketing numbers, satisfy uniform laws…
In this short mostly expository note, we sketch a program for gauging fully extended topological field theories in 3 dimensions. One begins with the spherical fusion category with which one wants to do Levin-Wen or Turaev-Viro. One then…
The extent of mixed regions around convective zones is one of the biggest uncertainties in stellar evolution. 1D overshooting descriptions introduce a free parameter ($f_{ov}$) that is in general not well constrained from observations.…
Frankl--Pach and Erd\H{o}s conjectured that any $(d+1)$-uniform set family $\mathcal{F}\subseteq \binom{[n]}{d+1}$ with VC-dimension at most $d$ has size at most $\binom{n-1}{d}$ when $n$ is sufficiently large. Ahlswede and Khachatrian…
A family of $k$ point sets in $d$ dimensions is well-separated if the convex hulls of any two disjoint subfamilies can be separated by a hyperplane. Well-separation is a strong assumption that allows us to conclude that certain kinds of…
We prove that, for any $d$ linearly independent functions from some set into a $d$-dimensional vector space over any field, the family of zero sets of all non-trivial linear combination of these functions has VC-dimension and Littlestone…
We relate various concepts of fractal dimension of the limiting set C in fractal percolation to the dimensions of the set consisting of connected components larger than one point and its complement in C (the "dust"). In two dimensions, we…
In the paper we consider convex cones in infinite-dimensional real vector spaces which are endowed with no topology. The main purpose is to study an internal geometric structure of convex cones and to obtain an analytical description of…
We show that the bicovariant first order differential calculi on a factorisable semisimple quantum group are in 1-1 correspondence with irreducible representations $V$ of the quantum group enveloping algebra. The corresponding calculus is…
This work studies embedding of arbitrary VC classes in well-behaved VC classes, focusing particularly on extremal classes. Our main result expresses an impossibility: such embeddings necessarily require a significant increase in dimension.…
Let $G$ be a connected, simply connected three-dimensional Lie group (unimodular or non-unimodular) equipped with a left-invariant (Riemannian or Lorentzian) metric $g$. By definition, the isometry group $\mathrm{Isom}(G, g)$ contains $G$…
Vector Fitting (VF) is a popular method of constructing rational approximants that provides a least squares fit to frequency response measurements. In an earlier work, we provided an analysis of VF for scalar-valued rational functions and…
We estimate a Box-counting dimension of fractal surfaces which are generated by iterated function systems with a vertical contraction factor function on an arbitrary data set over rectangular grids and can express well a lot of natural…
We show that the core reasons that complex and hypercomplex valued neural networks offer improvements over their real-valued counterparts is the weight sharing mechanism and treating multidimensional data as a single entity. Their algebra…
Consider all the level sets of a real function. We can group these level sets according to their Hausdorff dimensions. We show that the Hausdorff dimension of the collection of all level sets of a given Hausdorff dimension can be…
We study a family of invariants of compact metric spaces that combines the Curvature Sets defined by Gromov in the 1980s with Vietoris-Rips Persistent Homology. For given integers $k\geq 0$ and $n\geq 1$ we consider the dimension $k$…
High-dimensional expanders are a generalization of the notion of expander graphs to simplicial complexes and give rise to a variety of applications in computer science and other fields. We provide a general tool to construct families of…
Geometric deep learning has attracted significant attention in recent years, in part due to the availability of exotic data types for which traditional neural network architectures are not well suited. Our goal in this paper is to…