Study of singular curves in a specific type of hyperbolic distribution.
problem Characterizing singular curves in hyperbolic (4,7)-distributions. method Introduced hyperbolic (4,7)-distributions of type C3, described singular curves via prolongations. result Completely described singular curves for hyperbolic (4,7)-distributions of type C3. Model-based clustering imposes a finite mixture modelling structure on data for clustering. Finite mixture models assume that the population is a convex combination of a finite number of densities, the distribution within each population is a basic assumption of each particular model. Among all distributions that have …
In this study, a numerical quadrature for the generalized inverse Gaussian distribution is derived from the Gauss-Hermite quadrature by exploiting its relationship with the normal distribution. The proposed quadrature is not Gaussian, but it exactly integrates the polynomials of both positive and negative orders. Using…
Hyperbolic space is a geometry that is known to be well-suited for representation learning of data with an underlying hierarchical structure. In this paper, we present a novel hyperbolic distribution called \textit{pseudo-hyperbolic Gaussian}, a Gaussian-like distribution on hyperbolic space whose density can be evalua…
Study on lengths of random multicurves on hyperbolic surfaces.
problem Distribution of lengths of random multicurves on closed hyperbolic surfaces.
method Using Margulis' thesis and Mirzakhani's equidistribution theorem for horospheres.
result Distribution of lengths admits a polynomial density, with coefficients expressible in terms of intersection numbers of psi-classes.
Study spectral distribution of twisted Laplacian on high genus hyperbolic surfaces.
problem Estimating spectral distribution of twisted Laplacian on hyperbolic surfaces.
method Estimate spectral distribution by supremum norm of harmonic form; show small supremum norm for high genus surfaces; prove uniform Weyl law.
result Prove uniform Weyl law for real parts of spectrum on high genus hyperbolic surfaces.
Geometric approach solves maximum likelihood for Cauchy-like distributions.
problem Estimating center and scatter robustly from heavy-tailed data.
method Geodesic convexity and symmetry spaces of noncompact type.
result Efficient numerical solution for robust estimates of location and spread.
Study explores geometric structure and prior for beta-logistic distribution.
problem Understanding the geometric structure and prior distributions of the beta-logistic distribution.
method Exploring dual geometric structure and uncovering α-parallel prior. result The beta-logistic distribution admits an α-parallel prior for any real number α. Proves weak convergence equals mean convergence in GGC.
problem Proving convergence in GGC distributions.
method Using generalized gamma convolution (GGC) and expected utility maximization.
result Weak convergence implies mean convergence in GGC.
A generic geodesic on a finite area, hyperbolic 2-orbifold exhibits an infinite sequence of penetrations into a neighborhood of a cone singularity, so that the sequence of depths of maximal penetration has a limiting distribution. The distribution function is the same for all such surfaces and is described by a fairly …
Improves few-shot learning for hierarchical data using hyperbolic space.
problem Few-shot class-incremental learning for hierarchical data.
method Contrastive learning in hyperbolic space, Poincaré ball model, hyperbolic contrastive loss, maximum entropy distribution.
result Effective improvement of coarse and fine class accuracies in few-shot conditions.
The study explores dilating set properties across Euclidean and hyperbolic geometries.
problem Distributional properties of dilating sets under projection.
method Covering maps and unit tangent bundles, focusing on manifolds of constant curvature.
result Established a precise asymptotic expansion for averages along expanding translates of homogeneous curves in hyperbolic surfaces.
Statistical hyperbolicity proven for Teichmüller space.
problem Harmonic measures from random walks on mapping class groups.
method Proving statistical hyperbolicity using Teichmüller metric.
result Teichmüller space is statistically hyperbolic for certain harmonic measures.
This paper solves a complex differential relation using a novel 'avoidance trick'.
problem Classifying tangent distributions satisfying non-involutivity conditions.
method Convex integration with an 'avoidance trick'.
result First example of a differential relation that is ample in some directions but not all.
New GP-based MIL method using Hyperbolic Secant distribution.
problem Probabilistic MIL for medical image analysis.
method Formulate VGPMIL using Pólya-Gamma random variables.
result Competitive or superior predictive performance and efficiency.
The study reveals a persistent bias in the distribution of holonomy on compact hyperbolic 3-manifolds.
problem The distribution of holonomy on compact hyperbolic 3-manifolds is not uniformly distributed.
method An asymptotic count of closed geodesics by their length and holonomy, and analysis of spectral parameters.
result A normalized, smoothed bias count of holonomy is distributed according to a probability distribution, controlled by the number of zero spectral parameters.
The study investigates how branch points affect the shape and mechanics of hyperbolic surfaces.
problem Understanding the role of branch points in the shape and mechanics of hyperbolic surfaces.
method Developed a discrete differential geometric (DDG) approach to study deformations of hyperbolic objects with distributed branch points.
result Branch points influence the overall morphology of hyperbolic surfaces without concentrating energy, leading to sub-exponential growth in maximum curvature.
The study finds many tight contact structures on hyperbolic 3-spheres.
problem Finding tight contact structures on hyperbolic 3-spheres.
method Constructing hyperbolic homology 3-spheres and analyzing their tight contact structures.
result Produces hyperbolic homology 3-spheres with multiple distinct tight contact structures.
Inverts operator on hyperbolic surfaces, constructing invariant distributions.
problem Constructing explicit inversion formula for X-ray normal operator.
method First, inversion formula for attenuated normal operator on Poincaré disk and closed hyperbolic surfaces. Then, explicit construction of invariant distributions.
result Explicit construction of invariant distributions with prescribed pushforward.
New normalizing flows in hyperbolic space improve posterior modeling for hierarchical data.
problem Limited flexibility of existing normalizing flows in Euclidean space for hierarchical data.
method Elevated normalizing flows to hyperbolic spaces using coupling transforms and Wrapped Hyperboloid Coupling.
result Improved performance on density estimation and hierarchical graph data.
The paper shows how to construct non-Gaussian Martingales using hyperbolic diffusion.
problem The challenge of modeling extreme financial events.
method Constructing Martingale processes with Cauchy distribution in the large volatility limit.
result Financial justification for using non-Gaussian distributions in modeling extreme events.
Develops a new model to track financial market interconnectedness over time.
problem Investigating time-varying financial market interconnectedness.
method Hidden Markov graphical model with state-dependent generalized hyperbolic distributions.
result Identifies different degrees of network connectivity of returns over time.
The paper classifies 3D contact partially hyperbolic diffeomorphisms.
problem Classifying contact partially hyperbolic diffeomorphisms in 3D.
method Smooth classification, conjugation to known flows or automorphisms, use of invariant distributions.
result Classification up to finite quotient or power, conjugation to known structures.
APo-VAE generates text in hyperbolic space for better hierarchical representation.
problem Lack of hierarchical structure in Euclidean embeddings for natural language.
method Adversarial Poincare Variational Autoencoder (APo-VAE) in hyperbolic latent space.
result APo-VAE outperforms Euclidean VAEs in capturing latent language hierarchies.
Paper shows how to evenly distribute intersections in hyperbolic spaces.
problem Equidistribution of intersections in hyperbolic manifolds.
method Properly immersed totally geodesic submanifolds, hyperbolic volume measure.
result Effective equidistribution of intersection points as submanifolds grow.
Robust clustering of high-dimensional data is an important topic because clusters in real datasets are often heavy-tailed and/or asymmetric. Traditional approaches to model-based clustering often fail for high dimensional data, e.g., due to the number of free covariance parameters. A parametrization of the component sc…
In this note we study the distribution of real inflection points among the ovals of a real non-singular hyperbolic curve of even degree. Using Hilbert's method we show that for any integers d and r such that 4≤r≤2d2−2d, there is a non-singular hyperbolic curve of degree 2d in R2 with exactl…
Each free homotopy class of directed closed curves on a surface with boundary can be described by a cyclic reduced word in the generators of the fundamental group and their inverses. The word length is the number of letters of the cyclic word. If the surface has a hyperbolic metric with geodesic boundary, the geometric…
We describe a hyperbolic version of the Ambartzumian-Pleijel identity. We use this identity to prove the hyperbolic Crofton formula and the hyperbolic isoperimetric inequality. This identity also provides a way to compute the chord length distribution for an ideal polygon in the hyperbolic plane. The analogous results …
Let S=Γ\H be a hyperbolic surface of finite topological type, such that the Fuchsian group Γ≤PSL2(R) is non-elementary, and consider any generating set S of Γ. When sampling by an n-step random walk in π1(S)≅Γ with each step given by an element…
Asymptotics for equidistribution of circles on hyperbolic surfaces.
problem Equidistribution of circles on hyperbolic surfaces.
method Spectral method and statistical limit theorems.
result Precise asymptotics for the rate of equidistribution of circles.
BIGUE algorithm provides credible intervals for hyperbolic network embeddings.
problem Uncertainty in hyperbolic network embeddings.
method Markov chain Monte Carlo (MCMC) algorithm for Bayesian hyperbolic random graph model.
result Samples from the posterior distribution provide credible intervals for hyperbolic coordinates and network properties.
Concrete distribution properties examined on simplex.
problem Properties of Concrete distribution on simplex.
method Reflection and location-scale transformation of uniform distribution; explicit parameterization to Poincaré half-space.
result Fisher information and information metric are hyperbolic space; Fisher-Rao geodesic distance computed.
New hyperbolic sliced-Wasserstein distances derived for efficient comparison.
problem Efficient comparison of distributions in hyperbolic spaces.
method Projections on geodesics or horospheres to derive novel sliced-Wasserstein distances.
result Novel hyperbolic sliced-Wasserstein distances are more computationally efficient.
We give an effective estimate for the totally ramified value number of the hyperbolic Gauss maps of complete flat fronts in the hyperbolic three-space. As a corollary, we give the upper bound of the number of exceptional values of them for some topological cases. Moreover, we obtain some new examples for this class.
We study the distribution of resonances for geometrically finite hyperbolic surfaces of infinite area by countting resonances numerically. The resonances are computed as zeros of the Selberg zeta function, using an algorithm for computation of the zeta function for Schottky groups. Our particular focus is on three aspe…
The study of isospectral surfaces in Euclidean and hyperbolic geometries.
problem Existence of non-isometric surfaces with identical chord length distributions.
method Construction of isospectral pairs of hyperbolic surfaces without common covers.
result Found isospectral pairs of hyperbolic surfaces with no common cover.
This paper improves probabilistic latent models on hyperbolic spaces.
problem Uncertainty in predictions due to geodesics crossing low-data regions.
method Augmenting hyperbolic manifold with a pullback metric for probabilistic pullback metrics.
result Geodesics on pullback metric respect both geometry and data distribution, reducing uncertainty.
We investigate relation between Dehn fillings and commensurability of hyperbolic 3-manifolds. The set consisting of the commensurability classes of hyperbolic 3-manifolds admits the quotient topology induced by the geometric topology. We show that this quotient space satisfies some separation axioms. Roughly speaking, …
In 3-dimensional hyperbolic geometry, the classical Schlafli formula expresses the variation of the volume of a hyperbolic polyhedron in terms of the length of its edges and of the variation of its dihedral angles. We prove a similar formula for the variation of the volume of the convex core of a geometrically finite h…
We give the best possible upper bound on the number of exceptional values and the totally ramified value number of the hyperbolic Gauss map for pseudo-algebraic constant mean curvature one surfaces in the hyperbolic three-space and some partial results on the Osserman problem for algebraic case. Moreover, we study the …
The study quantifies distances between certain hyperbolic surfaces and bounds their number.
problem Quantifying distances and bounding numbers of hyperbolic surfaces.
method Parametrization of Teichmüller space by length functions.
result Two coverings of arithmetic surfaces are $P( rac{1}{g})$ apart, with P depending only on the surface. HLoOP detects outliers in hyperbolic 2-space.
problem Detecting local outliers in hyperbolic 2-space.
method Combines nearest neighbor finding and probabilistic scoring in hyperbolic space.
result Promising results on WordNet dataset.
New method calculates winding of geodesics on surfaces.
problem Understanding the distribution of geodesics on surfaces.
method Introducing a new construction of winding numbers for geodesics on cusped hyperbolic orbifolds.
result Winding numbers can be expressed by Rademacher symbols for various arithmetic families of surfaces.
Cohomology fractals illustrate complex 3-manifold properties.
problem Visualizing complex cohomology classes in hyperbolic 3-manifolds.
method Ray-tracing cohomology fractals and proving their distribution.
result Cohomology fractals converge to a distribution on the sphere at infinity.
The main goal of this article is to understand how the length spectrum of a random surface depends on its genus. Here a random surface means a surface obtained by randomly gluing together an even number of triangles carrying a fixed metric. Given suitable restrictions on the genus of the surface, we consider the number…
Study on length spectrum of random hyperbolic 3-manifolds.
problem Understanding the length spectrum of random hyperbolic 3-manifolds.
method Modeling random hyperbolic 3-manifolds using truncated tetrahedra and analyzing their length spectrum as volume tends to infinity.
result The length spectrum converges in distribution to a Poisson point process with a computable intensity λ as volume increases.
Study on hyperbolic polyhedra and their volume, proving finiteness of arithmetic groups.
problem Finiteness of arithmetic maximal reflection groups in hyperbolic polyhedra.
method Observation of volume distribution and recent work with M. Fraczyk and S. Hurtado.
result Proof of finiteness of arithmetic maximal reflection groups.