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.
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.
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 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.
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.
Paper extends causal inference to non-Euclidean data like images and distributions.
problem Causal inference for non-Euclidean data like images and distributions.
method Hilbert space embeddings, Fréchet mean estimation, nonparametric doubly-debiased causal inference.
result Validated approach for causal inference with continuous treatments on non-Euclidean data.
A non-Euclidean generalization of conditional expectation is introduced and characterized as the minimizer of expected intrinsic squared-distance from a manifold-valued target. The computational tractable formulation expresses the non-convex optimization problem as transformations of Euclidean conditional expectation. …
New algorithms optimize convex functions with high-order derivatives.
problem Optimizing convex functions with high-order derivatives under various norms.
method Developed a non-Euclidean inexact accelerated proximal point method using an inexact uniformly convex regularizer.
result Showed nearly optimal algorithms for high dimensions in the black-box oracle model for ℓp-settings and all q≥1. EF21-Muon optimizes deep learning with error feedback, improving efficiency and accuracy.
problem Lack of principled distributed frameworks for non-Euclidean LMO-based optimizers.
method Introduces EF21-Muon, a communication-efficient, non-Euclidean LMO-based optimizer with convergence guarantees.
result First efficient distributed implementation of non-Euclidean LMO-based optimizers, achieving up to 7x communication savings.
New graph convolution captures local features on non-Euclidean grids.
problem Capturing local features on irregular, coarse non-Euclidean grids.
method Low-rank learnable local filters in graph convolutions.
result Proves more expressive than previous spectral graph convolution methods.
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 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 method estimates causal effects in complex spaces using topological structures.
problem Challenges in estimating causal effects in non-Euclidean spaces.
method Developed a topological causal inference framework using power-weighted silhouette functions of persistence diagrams.
result Successfully quantifies topological treatment effects across various complex outcomes.
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.
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.
We obtain the parametric equations of all biharmonic Legendre curves and Hopf cylinders in the 3-dimensional unit sphere endowed with the modified Sasakian structure defined by Tanno.
Proposes IIKL for preserving geometric properties of non-Euclidean data.
problem Loss of geometric information in non-Euclidean data representation.
method IIKL method builds Riemannian manifold and isometrically induces metric.
result Preserves geometric structure of original data in 3D and high-dimensional datasets.
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.
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.
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.
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.
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.
Develops Gaussian processes on non-Euclidean spaces with symmetries.
problem Invariance to symmetries in non-Euclidean spaces.
method Constructive techniques for stationary Gaussian processes on compact and non-compact spaces.
result Makes non-Euclidean Gaussian processes compatible with standard software.
Novel Fréchet regression method handles errors-in-variables with low-rank covariates.
problem Regression with noisy and limited covariate data.
method Combines global Fréchet regression and principal component regression for low-rank structure.
result Improved efficiency and accuracy in high-dimensional and noisy data settings.
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.
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.
The study explores properties and mutations in oriented matroids, proving new results on Euclidean and non-Euclidean structures.
problem Investigating the Euclidean and non-Euclidean properties of oriented matroids.
method Analyzing the minimum number of mutations, using lexicographic extensions, and mutation-flips to prove properties.
result For rank 4 uniform oriented matroids, the minimum number of mutations adjacent to an element is at most 3.
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.
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.
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.
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. 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.
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.
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.
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.
New method for learning with non-Euclidean data using decomposable kernels.
problem Difficulty in using classical kernels for non-Euclidean data.
method Reproducing kernel Krein space (RKKS) methods for kernels that admit a positive decomposition.
result Invariant kernels can be used for learning in non-Euclidean spaces.
This work tackles regression on non-Euclidean spaces, specifically positive-definite matrices with the Bures-Wasserstein metric.
problem Regression on non-Euclidean spaces, specifically positive-definite matrices with the Bures-Wasserstein metric.
method Developed a sufficient condition for the existence of a minimizer of the conditional barycenter problem, characterized the optimization landscape, and developed a projection-free algorithm for approximate computation of first-order stationary points.
result The objective is free of local maxima under the sufficient condition, and the algorithm enables the use of stochastic Riemannian optimization methods for large-scale setups.
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.
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.
New framework improves robustness of implicit neural networks.
problem Ill-posedness and convergence instability in implicit neural networks.
method NEMON framework based on contraction theory for ℓ∞ norm, including well-posedness condition, average iteration, and input-output Lipschitz constant regularization. result Improved accuracy and robustness of implicit models with smaller input-output Lipschitz bounds.
DFNNs predict non-Euclidean responses from Euclidean predictors.
problem Regression with non-Euclidean responses.
method Deep Fréchet neural networks (DFNNs) approximating conditional Fréchet means.
result DFNNs consistently outperform existing methods in empirical studies.