Study classifies graphs in Euclidean and non-Euclidean spaces with specific curvature conditions.
problem Classifying graphs with prescribed curvature in various spaces.
method Proves rigidity and classification results for graphs in Riemannian manifolds, focusing on R2 and R3. result Provides general splitting theorems for graphs in these settings.
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.
This study bridges the gap between spatial and spectral GNNs.
problem Lack of direct comparison and cross-reference of existing GNNs.
method Systematically categorizes and examines GNNs into spatial and spectral domains.
result Establishes a strong relationship between spatial and spectral GNNs.
Paper introduces signal processing on cell complexes.
problem Processing signals on non-Euclidean domains.
method Signal processing on abstract regular cell complexes.
result Hodge Laplacians for cell complexes enable convolutional filters.
Optimizes two-sample tests for non-Euclidean domains using spectral regularization.
problem Optimizing two-sample tests for non-Euclidean domains.
method Spectral regularization of MMD test to achieve minimax optimality.
result Proposes a spectral regularization method that improves test optimality.
Mean curvature flow converges to a translating soliton with prescribed contact angle.
problem Mean curvature flow with contact angle constraints in non-Euclidean settings.
method Existence proof using translating solitons and bounds on convexity and Ricci curvature.
result Graphical solutions converge to a translating soliton as time goes to infinity.
UNOT solves optimal transport problems efficiently using neural networks.
problem Computational expense in solving optimal transport problems.
method UNOT (Universal Neural Optimal Transport) uses Fourier Neural Operators to predict OT distances and plans accurately and efficiently.
result UNOT achieves up to 7.4x speedup over the Sinkhorn algorithm while maintaining accuracy.
We propose a novel Bayesian nonparametric method to learn translation-invariant relationships on non-Euclidean domains. The resulting graph convolutional Gaussian processes can be applied to problems in machine learning for which the input observations are functions with domains on general graphs. The structure of thes…
Stability is a key aspect of data analysis. In many applications, the natural notion of stability is geometric, as illustrated for example in computer vision. Scattering transforms construct deep convolutional representations which are certified stable to input deformations. This stability to deformations can be interp…
The paper proves a Neumann eigenvalue sum inequality in non-Euclidean space forms.
problem Proving an inequality involving Neumann eigenvalues in non-Euclidean spaces.
method Analyzing space forms with constant curvature and using geodesic balls.
result Proves a conjecture about Neumann eigenvalues in non-Euclidean spaces.
Virtual reality explores non-Euclidean Sol geometry.
problem Exploring non-Euclidean geometries in virtual reality.
method Developed a VR software for Sol geometry.
result Demonstrated the feasibility of non-Euclidean geometry in VR.
Accelerates signature kernel computation for sequences.
problem Severe computational bottleneck in computing signature kernel.
method Random Fourier features to accelerate signature kernel computation.
result Uniform approximation guarantees for unbiased estimator with linear computation time.
Gaussian processes adapted for non-Euclidean spaces enhance decision-making.
problem Applying Gaussian processes in non-Euclidean spaces.
method Developed pathwise conditioning and Gaussian process models over non-Euclidean spaces.
result Efficient Gaussian process models for non-Euclidean spaces.
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.
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.
This work presents a reformulation of the recently proposed Wasserstein autoencoder framework on a non-Euclidean manifold, the Poincaré ball model of the hyperbolic space. By assuming the latent space to be hyperbolic, we can use its intrinsic hierarchy to impose structure on the learned latent space representations. W…
New families of non-tiling domains satisfy Pólya's conjecture.
problem Finding non-tiling domains that satisfy Pólya's conjecture.
method Analyzing partitioning and eigenvalue orders of domains.
result Existence of families of non-tiling domains satisfying Pólya's conjecture.
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.
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.
The paper introduces novel Gaussian process models for vector-valued signals on manifolds.
problem Modeling vector-valued signals on non-Euclidean domains, especially for applications like wind speeds.
method Intrinsically defined Gaussian vector fields on manifolds, accounting for manifold geometry.
result Gaussian vector fields provide more refined inductive biases than extrinsic fields.
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.
Graph neural networks improve predictions on graph data.
problem Complex non-Euclidean graph data limits traditional machine learning methods.
method Graph neural networks for node-level predictions.
result Improved handling of large-scale and time-dynamic graphs.
Neuc-MDS extends MDS for non-Euclidean data.
problem Limitations of classical MDS with non-Euclidean data.
method Generalizes inner product to symmetric bilinear forms, optimizes eigenvalues of dissimilarity Gram matrix.
result Optimizes STRESS for non-Euclidean data.
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.
Algorithm improves SVM classification in non-Euclidean spaces.
problem Limitations of traditional SVM in non-Euclidean spaces.
method Covariance-adjusted SVM using Cholesky Decomposition.
result Cholesky-SVM outperforms traditional SVM in non-Euclidean spaces.
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.
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.
This paper considers the problem of low-dimensional visualisation of very high dimensional information sources for the purpose of situation awareness in the maritime environment. In response to the requirement for human decision support aids to reduce information overload (and specifically, data amenable to inter-point…
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.
This paper deals with various topics in analysis on hyperbolic spaces. It surveys some recent progress in non-Euclidean Fourier Analysis and proves some new results for the geodesic Radon transform on hyperbolic spaces.
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.
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.
This paper tightens the generalization error bound for graph embedding in non-Euclidean spaces.
problem High generalization error in non-Euclidean graph embedding, preventing practical applications.
method Novel upper bound of graph embedding's generalization error using local Rademacher complexity.
result The new bound is tighter and faster, allowing better performance in non-Euclidean spaces.
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.
For each geometrically finite 2-dimensional non-Euclidean crystallographic group (NEC group), we compute the cohomology groups. In the case where the group is a Fuchsian group, we also determine the ring structure of the cohomology.
In this paper we demonstrate how the geometrically motivated algorithm to determine whether a two generator real Mobius group acting on the Poincare plane is or is not discrete can be interpreted as a non-Euclidean Euclidean algorithm. That is, the algorithm can be viewed as an application of the Euclidean division alg…
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.
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.
New flows model distributions on Riemannian manifolds without domain knowledge.
problem Limited modeling of distributions on Riemannian manifolds.
method Riemannian convex potential maps using optimal transport.
result These flows can model standard distributions on spheres and tori.
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.
New optimization method combines gradient clipping and non-Euclidean smoothness.
problem Improving optimization in non-Euclidean spaces for machine learning.
method Hybrid of steepest descent and conditional gradient, incorporating weight decay.
result Achieves optimal convergence rate and demonstrates effectiveness in deep learning.
Imaging-based early diagnosis of Alzheimer Disease (AD) has become an effective approach, especially by using nuclear medicine imaging techniques such as Positron Emission Topography (PET). In various literature it has been found that PET images can be better modeled as signals (e.g. uptake of florbetapir) defined on a…
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. 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.
This work proposes hyperbolic deep convolutional neural networks for better pattern recognition.
problem The limitations of Euclidean deep convolutional neural networks in capturing intricate patterns.
method Developed Hyperbolic DCNN based on Poincaré Disc, analyzing expansive convolution in non-Euclidean space.
result Hyperbolic convolutional architecture outperforms Euclidean ones in pattern recognition tasks.
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.
Develops statistical methods for rates of change on Riemannian manifolds.
problem Statistical inference for rates of change in spatial processes over non-Euclidean domains.
method Formalizes smoothness and constructs differential processes for Riemannian manifolds, derives conditions for kernel existence, and develops predictive inference.
result Validates theoretical findings through simulation experiments for derivatives over polyhedral meshes.
Researchers develop neural networks for manifold data with a convergence rate.
problem Analyzing high-dimensional data on non-Euclidean domains.
method Constructing manifold neural networks using spectral decomposition of the Laplace Beltrami operator.
result Established a rate of convergence for the neural network scheme that depends on intrinsic manifold dimension.