Related papers: Hyperbolic Voronoi diagrams made easy

We introduce hyperbolic attention networks to endow neural networks with enough capacity to match the complexity of data with hierarchical and power-law structure. A few recent approaches have successfully demonstrated the benefits of…

Data representation in non-Euclidean spaces has proven effective for capturing hierarchical and complex relationships in real-world datasets. Hyperbolic spaces, in particular, provide efficient embeddings for hierarchical structures. This…

We numerically study quasiperiodic normally hyperbolic attracting invariant circles that appear for certain parameter values in a family of three-dimensional Henon-like maps. These parameter values make up contour segments in the parameter…

Decision trees and models that use them as primitives are workhorses of machine learning in Euclidean spaces. Recent work has further extended these models to the Lorentz model of hyperbolic space by replacing axis-parallel hyperplanes with…

Recently, there has been a rising surge of momentum for deep representation learning in hyperbolic spaces due to theirhigh capacity of modeling data like knowledge graphs or synonym hierarchies, possessing hierarchical structure. We refer…

For image segmentation, the current standard is to perform pixel-level optimization and inference in Euclidean output embedding spaces through linear hyperplanes. In this work, we show that hyperbolic manifolds provide a valuable…

In this work we present a useful way to introduce the octonionic projective and hyperbolic plane through the use of Veronese vectors. Then we focus on their relation with the exceptional Jordan algebra and show that the Veronese vectors are…

We consider the quasiconformal dilatation of projective transformations of the real projective plane. For non-affine transformations, the contour lines of dilatation form a hyperbolic pencil of circles, and these are the only circles that…

Irregular arrangements of vesicles filled with excitable and precipitating chemical systems are imitated by Voronoi automata --- finite-state machines defined on a planar Voronoi diagram. Every Voronoi cell takes four states: resting,…

As an increasingly powerful technique in integrated photonics, inverse design uses optimization algorithms to automatically create compact, high-performance photonic structures, often yielding non-intuitive layouts far more compact than…

We construct and study the ideal Poisson--Voronoi tessellation of the product of two hyperbolic planes $\mathbb{H}_{2}\times \mathbb{H}_{2}$ endowed with the $L^{1}$ norm. We prove that its law is invariant under all isometries of this…

A Voronoi diagram partitions the plane into convex cells, each containing the points closest to a single generator. Given such a tessellation, the inverse Voronoi problem seeks the generator set \( S \) that produced it. Our algorithm…

In this article, we investigate vacuum leakage detection problems in composite manufacturing. Our approach uses Voronoi diagrams, a well-known structure in discrete geometry. The Voronoi diagram of the vacuum connection positions partitions…

For piecewise-linear maps the stable and unstable manifolds of hyperbolic periodic solutions are themselves piecewise-linear. Hence compact subsets of these manifolds can be represented using polytopes (i.e. polygons, in the case of…

In this paper, we prove a limit set intersection theorem in relatively hyperbolic groups. Our approach is based on a study of dynamical quasiconvexity of relatively quasiconvex subgroups. Using dynamical quasiconvexity, many well-known…

We construct countable Markov partitions for non-uniformly hyperbolic diffeomorphisms on compact manifolds of any dimension, extending earlier work of O. Sarig for surfaces. These partitions allow us to obtain symbolic coding on invariant…

Let $S$ be a set of $n$ points in $\mathbb{R}^2$. Our goal is to preprocess $S$ to efficiently compute the smallest enclosing disk of the points in $S$ that lie inside an axis-aligned query rectangle. Previous data structures for this…

Many physical systems can be studied as collections of particles embedded in space, evolving through deterministic evolution equations. Natural questions arise concerning how to characterize these arrangements - are they ordered or…

Given a topological cell decomposition of a closed surface equipped with edge weights, we consider the Dirichlet energy of any geodesic realization of the 1-skeleton graph to a hyperbolic surface. By minimizing the energy over all possible…

Orbifolds are a modern mathematical concept that arises in the research of hyperbolic geometry with applications in computer graphics and visualization. In this paper, we make use of rooms with mirrors as the visual metaphor for orbifolds.…