This foreword discusses the contributions of Bolyai, Gauss, and Lobachevsky to non-Euclidean geometry.
problem The development of non-Euclidean geometries by Bolyai, Gauss, and Lobachevsky.
method Historical review of the contributions of these mathematicians.
result The foundational work on non-Euclidean geometries by Bolyai, Gauss, and Lobachevsky.
Study uses crochet to visualize non-Euclidean geometry.
problem Understanding non-Euclidean surfaces through physical models.
method Parametrization of crochet models to represent Lobachevskian surface.
result Crochet models reflect non-Euclidean geometry characteristics.
The study extends inscription problems to non-Euclidean geometries.
problem Generalizing inscription problems to non-Euclidean geometries.
method Symplectic and Riemannian geometry techniques.
result Proved generalized inscription theorems for hyperbolic and spherical surfaces.
Study of pulleys and gears in spherical and hyperbolic geometries.
problem Understanding mechanical systems in non-Euclidean spaces.
method Analysis of pulley and gear systems in spherical and hyperbolic geometries.
result Similar laws governing movement in non-Euclidean geometries.
The author suggests using non-Euclidean geometry for psychometric models.
problem Current psychometric models lack geometric insights.
method Illustrates how non-Euclidean geometry can be applied to psychometrics.
result Geometric concepts may improve psychometric model understanding.
These lecture notes are based on [arXiv: math/0702714, 0907.4469, 0907.4470]. We introduce and study basic aspects of non-Euclidean geometries from a coordinate-free viewpoint.
Non-Euclidean BPM extends optimization theory to non-Euclidean norms.
problem Extending BPM's convergence theory to non-Euclidean norms.
method Iteratively minimizing over norm balls in non-Euclidean geometry.
result Most BPM guarantees carry over to non-Euclidean norms.
Extends illumination bodies to non-Euclidean spaces and proves their volume derivative defines surface area.
problem Defining surface area in non-Euclidean geometries.
method Generalizes illumination bodies to Riemannian spaces of constant curvature and projective Finsler geometries, proving their volume derivative defines surface area.
result Derivative of volume of illumination bodies defines surface area in non-Euclidean geometries.
New samplers minimize KL divergence for constrained and non-Euclidean geometries.
problem Efficient sampling from constrained and non-Euclidean distributions.
method Stein Variational Mirror Descent and Mirrored Stein Variational Gradient Descent.
result New samplers converge more rapidly and accurately than prior methods.
We describe our initial explorations in simulating non-euclidean geometries in virtual reality. Our simulations of three-dimensional hyperbolic space are available at http://h3.hypernom.com.
GNPs learn operators on non-Euclidean geometries using neural networks.
problem Learning operators on complex geometries like manifolds.
method Geometric Neural Operators (GNPs) that incorporate geometric properties.
result GNPs can estimate metrics, solve PDEs, and learn LB operators on manifolds.
We study surfaces with decorations and prove uniformization in non-Euclidean geometries.
problem Discrete conformal equivalence in non-Euclidean geometries.
method Variational principle and continuous deformation.
result One master theory of discrete conformal equivalence across different geometries.
Virtual reality brings non-Euclidean geometry to life.
problem Understanding non-Euclidean geometry is challenging.
method Interactive visualizations in virtual reality.
result Users can experience non-Euclidean geometry firsthand.
Ray-marching method visualizes 8 Thurston geometries in real-time.
problem Accurately rendering and visualizing Thurston geometries in real-time.
method Ray-marching algorithms with theoretical framework for non-Euclidean geometries.
result Accurate interactive real-time views of Thurston geometries achieved.
We describe our initial explorations in simulating non-euclidean geometries in virtual reality. Our simulation of the product of two-dimensional hyperbolic space with one-dimensional euclidean space is available at http://h2xe.hypernom.com.
In two papers titled "On the so-called non-Euclidean geometry", I and II, Felix Klein proposed a construction of the spaces of constant curvature -1, 0 and and 1 (that is, hyperbolic, Euclidean and spherical geometry) within the realm of projective geometry. Klein's work was inspired by ideas of Cayley who derived the …
Paper improves SOMs for non-Euclidean data modeling.
problem Traditional SOMs assume Euclidean data, limiting their applicability.
method Introduces topology-related extensions to traditional SOM algorithm.
result Improves SOMs for non-Euclidean data, enhancing data modeling.
Develops a method to define and characterize geodesics on hyperbolic surfaces.
problem Characterizing closed geodesics on hyperbolic surfaces without self-intersection.
method Constructive definition of the Goldman bracket using closed geodesics.
result Algebraic characterization of geodesics on hyperbolic surfaces.
The edge of torn elastic sheets and growing leaves often form a hierarchical buckling pattern. Within non-Euclidean plate theory this complex morphology can be understood as low bending energy isometric immersions of hyperbolic Riemannian metrics. With this motivation we study the isometric immersion problem in strip a…
New research shows hyperbolic embeddings are useful for global consistency tasks in graphs.
problem The usefulness of hyperbolic representations in graph learning tasks.
method Computed hyperbolic embeddings for node classification and link prediction tasks, addressing optimization issues at zero curvature.
result Hyperbolic embeddings are more effective for tasks requiring global consistency, while Euclidean models are superior for other tasks.
Simplified proof of mass center system's uniqueness and generalized Pappus' theorem across Euclidean, spherical, and hyperbolic geometries.
problem Proving the uniqueness of the mass center system in non-Euclidean geometries and deriving a generalized Pappus' theorem.
method Revisiting and simplifying G.A. Galperin's proof, extending the mass center system to manifolds, and deriving a generalized Pappus' theorem.
result Unified and simpler proofs for Pappus' theorem in Euclidean, spherical, and hyperbolic geometries.
The paper explores centroids and static equilibrium points in non-Euclidean geometries.
problem Investigating centroids and static equilibrium points in spherical, hyperbolic, and normed spaces.
method Extending Gal'perin's work, the paper examines convex bodies in these spaces and analyzes the minimum number of equilibrium points.
result Every plane convex body in any of these spaces has at least four equilibrium points, and there are mono-monostatic convex bodies in 3D spherical, hyperbolic, and certain normed spaces.
Adaptive step-size improves optimization in complex geometries.
problem Optimizing functions with non-Euclidean geometries.
method Adaptive step-size strategy for optimization algorithms.
result Guaranteed convergence for Adaptive Conditional Gradient Descent.
This article presents virtual reality software designed to explore the Sol geometry. The simulation is available on 3-dimensional.space/sol.html
Piecewise flat approximations for curvature in Euclidean and non-Euclidean spaces.
problem Approximating local extrinsic curvature on discrete manifolds.
method Constructing discrete curvature forms on piecewise flat manifolds, using weighted sums of hinge angles.
result Converges to smooth curvature values as mesh refinement occurs, favorably comparing with other discrete approaches.
New method turns optimization algorithms into uniformly stable learning algorithms for non-Euclidean norms.
problem Non-Euclidean norms in binary classification problems.
method Black-box reduction method using uniformly convex regularizers.
result Achieves optimal statistical risk bounds on excess risk for non-Euclidean norms.
In this paper, we consider the problem of fast and efficient indexing techniques for sequences evolving in non-Euclidean spaces. This problem has several applications in the areas of human activity analysis, where there is a need to perform fast search, and recognition in very high dimensional spaces. The problem is ma…
This work closes the theory-practice gap for distributed optimization methods by introducing a new regularity condition.
problem Existing convergence conditions for distributed optimization methods are violated by nearly all kernels used in practice.
method Introduces Hessian relative uniform continuity (HRUC) to guarantee convergence under mild conditions.
result Derives convergence guarantees for mirror descent-based gradient tracking without restrictive assumptions.
The paper introduces a new geometric representation for data.
problem Representing tree-like data more effectively in non-Euclidean spaces.
method Develops a representation on a pseudo-Riemannian manifold of constant nonzero curvature.
result Provides closed-form expressions for distances and descent directions.
Gradient descent near stability threshold shows sharpness oscillations.
problem Understanding sharpness and stability in non-Euclidean norms during gradient descent.
method Interpreted EoS through Directional Smoothness, defined generalized sharpness for arbitrary norms.
result Non-Euclidean GD exhibits sharpness oscillations around the stability threshold.
Gradient descent near stability threshold exhibits sharpness oscillations.
problem Understanding sharpness behavior near stability threshold in non-Euclidean norms.
method Interpreted EoS through Directional Smoothness and generalized sharpness under arbitrary norms.
result Non-Euclidean GD with generalized sharpness shows sharpness oscillations near 2/η. This work analyzes Fréchet regression using comparison geometry, providing theoretical and practical insights.
problem Analyzing data on complex structures like manifolds and graphs.
method Theoretical analysis through comparison geometry, focusing on existence, uniqueness, and stability of the Fréchet mean.
result Key results on the existence, uniqueness, and stability of the Fréchet mean, along with statistical guarantees for nonparametric regression.
The goal of this paper is to introduce and study analogues of the Euclidean Funk and Hilbert metrics on open convex subsets Ω of hyperbolic or spherical spaces. At least at a formal level, there are striking similarities among the three cases: Euclidean, spherical and hyperbolic. We start by defining non-Euclidean an…
Paper accelerates Bayesian few-shot classification using mirror descent.
problem Non-conjugate inference in Bayesian few-shot classification.
method Integrates mirror descent-based variational inference into Gaussian process-based few-shot classification.
result Accelerated convergence and improved uncertainty quantification.
Conic singular sub-manifolds are Lipschitz Normally Embedded in compact non-Euclidean manifolds.
problem Understanding the Lipschitz geometry of conic singular sub-manifolds.
method Analyzing the metric properties of conic singular sub-manifolds in compact non-Euclidean manifolds.
result Connected conic singular sub-manifolds are Lipschitz Normally Embedded.
We describe a method of rendering real-time scenes in Nil geometry, and use this to give an expository account of some interesting geometric phenomena. You can play around with the simulation at www.3-dimensional.space/nil.html.
New Sliced-Wasserstein distances for non-Euclidean data.
problem Computational burden of Wasserstein distance on non-Euclidean manifolds.
method Derive Sliced-Wasserstein distances and flows on Cartan-Hadamard manifolds.
result General constructions and non-parametric schemes for minimizing new distances.
A new method learns manifold-valued latents without an encoder.
problem Distorting data with intrinsic non-Euclidean structure.
method Riemannian generative decoder that learns latents directly.
result Learned representations respect the prescribed geometry and capture intrinsic non-Euclidean structure.
Study on curvature measures in non-Euclidean spaces linked to Euclidean geometry.
problem Investigating curvature measures in spherical, hyperbolic, and de Sitter spaces.
method Establishing a unifying framework for curvature measures in real-analytic spaces of constant curvature.
result Floating bodies and duality in non-Euclidean spaces are connected to curvature measures in Euclidean space.
Generalizes Toponogov theorem to Alexandrov spaces.
problem Estimating curve length in non-Euclidean spaces.
method Generalization of Toponogov theorem.
result Proved the length of a curve in two-dimensional Alexandrov spaces.
A robust visual tracking system requires an object appearance model that is able to handle occlusion, pose, and illumination variations in the video stream. This can be difficult to accomplish when the model is trained using only a single image. In this paper, we first propose a tracking approach based on affine subspa…
Personalization of cardiac models involves the optimization of organ tissue properties that vary spatially over the non-Euclidean geometry model of the heart. To represent the high-dimensional (HD) unknown of tissue properties, most existing works rely on a low-dimensional (LD) partitioning of the geometrical model. Wh…
Information geometry applies concepts in differential geometry to probability and statistics and is especially useful for parameter estimation in exponential families where parameters are known to lie on a Riemannian manifold. Connections between the geometric properties of the induced manifold and statistical properti…
The seven non euclidean geometries of the Thurston's geometrization program are proved to originate naturally from singularization morphisms and versal deformations on euclidean 3-manifolds generated in the frame of the Langlands global program. The Poincare conjecture for a 3-manifold appears as a particular case of t…
Unified framework for non-Euclidean CPD under scalable stochastic mirror descent.
problem Handling non-Euclidean losses in tensor decomposition.
method Tensor fiber sampling strategy-based stochastic mirror descent.
result Global convergence to a stationary point under reasonable conditions.
Constant-curvature Riemannian manifolds (CCMs) have been shown to be ideal embedding spaces in many application domains, as their non-Euclidean geometry can naturally account for some relevant properties of data, like hierarchy and circularity. In this work, we introduce the CCM adversarial autoencoder (CCM-AAE), a pro…
Paper uses non-Euclidean analysis to classify brain structure variations.
problem Classifying joint variations in multi-object brain structures.
method Combines non-Euclidean statistics and non-parametric integrative analysis.
result Effective, robust, and interpretable joint structure found.
Theory models nonlinear soft tissue elasticity and remodeling using extended Finsler geometry.
problem Understanding and predicting the behavior of nonlinear soft tissues, especially in biologic contexts.
method Formulated a continuum mechanical theory incorporating extended Finsler geometry to describe the complex behaviors of fibrous soft solids.
result The model quantifies residual strains from growth, remodeling, and degradation, and predicts equilibrium configurations.