Related papers: Comparison theorems for hyperbolic type metrics
Representing data in hyperbolic space can effectively capture latent hierarchical relationships. With the goal of enabling accurate classification of points in hyperbolic space while respecting their hyperbolic geometry, we introduce…
We define and study hyperbolic extensions.
In the first part of this investigation, [Ha], we generalized a weighted distance function of [Li] and found necessary and sufficient conditions for it being a metric. In this paper some properties of this so-called M-relative metric are…
Singular and sectional hyperbolic sets are the objects of the extension of the classical Smale Hyperbolic Theory to flows having invariant sets with singularities accumulated by regular orbits within the set. It is by now well-known that…
We use a combinatorial approximation of the hyperbolic plane to investigate properties of hyperbolic geometry such as exponential growth of perimeter and area of disks, and the linear isoperimetric inequality. This calculations give a…
We study the growth of hyperbolic type distances in starlike domains. We derive estimates for various hyperbolic type distances and consider the asymptotic sharpness of the estimates.
A dimension reduction for the hyperbolic space is established. When points are far apart an embedding with bounded distortion into the hyperbolic plane is achieved.
This paper investigates the notion of learning user and item representations in non-Euclidean space. Specifically, we study the connection between metric learning in hyperbolic space and collaborative filtering by exploring Mobius…
An improved version of quasiinvariance property of the quasihyperbolic metric under M\"obius transformations of the unit ball in ${\mathbb R}^n, n \ge 2,$ is given. Next, a quasiinvariance property, sharp in a local sense, of the…
This is a tale describing the large scale geometry of Euclidean plane domains with their hyperbolic or quasihyperbolic distances. We prove that in any hyperbolic plane domain, hyperbolic and quasihyperbolic quasi-geodesics are the same…
We introduce several new functions that measure the distance between two points $x$ and $y$ in a domain $G\subsetneq\mathbb{R}^n$ by using the arithmetic or the logarithmic mean of the Euclidean distances from the points $x$ and $y$ to the…
In this paper, the versions of trigonometric functions of certain known inequalities for hyperbolic ones are proved, and then corresponding inequalities for means are presented.
The Hilbert metric on convex subsets of $\mathbb R^n$ has proven a rich notion and has been extensively studied. We propose here a generalization of this metric to subset of complex projective spaces and give examples of applications to…
Certain topics on polygons are extended from Euclidean to hyperbolic geometry. This first part deals with uniqueness and existence of cocyclic polygons with prescribed sidelengths. The non-Euclidean versions are more difficult due to the…
Let $K$ be a square in the plane and $\rho_K(x,y)$ be the hyperbolic distance between $x$, $y\in K$. Denote by $s_K(x,y)$ the triangular ratio metric in $K$; for $x\neq y$ the value of $s_K(x,y)$ equals the ratio of the Euclidean distance…
We suggest a new approach to the study of relatively hyperbolic groups based on relative isoperimetric inequalities. Various geometric, algebraic, and algorithmic properties are discussed.
We introduce the concept of a heterodimensional cycle of hyperbolic ergodic measures and a special type of them that we call rich. Within a partially hyperbolic context, we prove that if two measures are related by a rich heterodimensional…
Distance correlation is a recent extension of Pearson's correlation, that characterises general statistical independence between Euclidean-space-valued random variables, not only linear relations. This review delves into how and when…
A number of topics involving metrics and measures are discussed, including some of the special structure associated with ultrametrics.
We propose a hyperbolic set-to-set distance measure for computing dissimilarity between sets in hyperbolic space. While point-to-point distances in hyperbolic space effectively capture hierarchical relationships between data points, many…