We discuss some basic concepts of semi-Riemannian geometry in low-regularity situations. In particular, we compare the settings of (linear) distributional geometry in the sense of L. Schwartz and nonlinear distributional geometry in the sense of J.F. Colombeau.
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48 results for “distributional geometry”
Real AlphaBeta-Geometriesmath.DG
By a real alphabeta-geometry we mean a four-dimensional manifold M equipped with a neutral metric h such that (M,h) admits both an integrable distribution of alpha-planes and an integrable distribution of beta-planes. We obtain a local characterization of the metric when at least one of the distributions is parallel (i…
Study new -contact distributions and Lie systems.
problem Characterizing and understanding -contact distributions.
method Analyzing Goursat distributions and Lie systems.
result Characterized new types of -contact distributions.
Study on geometry of Dirichlet distributions using Fisher-Rao metric.
problem Understanding the geometry of Dirichlet distributions.
method Analysis of Fisher-Rao metric on Dirichlet distribution parameter space.
result Geodesic completeness and negative sectional curvature of the space.
Riemannian geometry for Compound Gaussian distributions: application to recursive change detectionstat.ML
New Riemannian geometry for Compound Gaussian distributions applied to efficient change detection.
problem Change detection in multivariate image times series.
method Developed a recursive approach based on Riemannian optimization.
result Optimal performance achieved with computational efficiency.
The notion of a parallelizable distribution has been introduced and investigated. A non-integrable parallelizable distribution carries a natural sub-Riemannian structure. The geometry of this structure has been studied from the bi-viewpoint of absolute parallelism geometry and sub-Riemannian geometry. Two remarkable li…
New KL-divergence for Gaussian distributions based on Wasserstein geometry.
problem Computing KL-divergence for Gaussian distributions efficiently.
method Introducing WKL-divergence based on Wasserstein geometry.
result WKL-divergence evaluates to squared distance between points for Dirac measures.
In this paper we describe the geometry of distributions by their symmetries, and present a simplified proof of the Frobenius theorem and some related corollaries. Then, we study the geometry of solutions of Gordon equation; A PDE which appears in differential geometry and relativistic field theory.
Whereas subriemannian geometry usually deals with smooth horizontal distributions, partially hyperbolic dynamical systems provide many examples of subriemannian geometries defined by non-smooth (namely, Hölder continuous) distributions. These distributions are of great significance for the behavior of the parent dynami…
The Fisher-Rao geometry is applied to elliptical distributions for optimization and classification.
problem Optimizing and classifying covariance matrices using geometric tools.
method Riemannian optimization and intrinsic Cramér-Rao bounds.
result Geometric tools enhance covariance matrix estimation and classification.
Develops k-contact geometry theory for field theories.
problem Analyse field theories using k-contact geometry.
method Distributions maximally non-integrable with k commuting Lie symmetries.
result Established k-contact distributions and their relationships.
We consider automorphisms of homogeneous parabolic geometries with a fixed point. Parabolic geometries carry the distinguished distributions and we study those automorphisms which enjoy natural actions on the distributions at the fixed points. We describe the sets of such automorphisms on homogeneous parabolic geometri…
Each sub-Riemannian geometry with bracket generating distribution enjoys a background structure determined by the distribution itself. At the same time, those geometries with constant sub-Riemannian symbols determine a unique Cartan connection leading to their principal invariants. We provide cohomological description …
We present the linearized metrizability problem in the context of parabolic geometries and subriemannian geometry, generalizing the metrizability problem in projective geometry studied by R. Liouville in 1889. We give a general method for linearizability and a classification of all cases with irreducible defining distr…
Optical geometriesmath.DG
Study of optical geometries with intrinsic torsion in general relativity.
problem Understanding null line distributions and their properties in Lorentzian manifolds.
method Investigation of intrinsic torsion and congruences of null curves, extending to generalized optical geometries.
result Characterization of conformal properties of null line distributions and congruences.
New framework uses geometry of embeddings to predict robustness.
problem Monitoring robustness in models without OOD labels.
method Constructs graphs from embeddings, measures spectral complexity and curvature.
result Representation geometry predicts robustness reliably.
A four-dimensional Walker geometry is a four-dimensional manifold M with a neutral metric g and a parallel distribution of totally null two-planes. This distribution has a natural characterization as a projective spinor field subject to a certain constraint. Spinors therefore provide a natural tool for studying Walker …
The paper uses geometric methods to classify medical data histograms.
problem Classifying medical data histograms for disease diagnosis.
method Information geometry of beta distributions for comparing and classifying histograms.
result Geometric tools, particularly negatively curved Fisher information, enable unique mean calculation and K-means classification.
Geometric aspects of rank-3 vector bundles over surfaces and 2-plane distributions on 5-manifoldsmath.DG
Geometric structures on surfaces relate to 2-plane distributions in 5D.
problem Understanding geometric properties of vector bundles and distributions.
method Study of horizontal 2-plane distributions on 5-manifolds.
result Established a connection between surface projective differential geometry and 2-plane distribution growth.
Study reveals CR structure of snake robot's geometry.
problem Understanding the kinematics and geometry of a snake robot.
method Analysis of (2,3,5) distributions and solving Cartan equivalence problem.
result Discovery of a CR structure with CR dimension 1 and real codimension 3.
A new distance metric for vMF distributions simplifies spherical data analysis.
problem Intractability of normalization constants and lack of suitable geometric metrics for comparing vMF distributions.
method Proposes a Wasserstein-like distance that decomposes vMF distribution discrepancies into angular and concentration components.
result The proposed distance metric induces a latent geometric structure on the space of non-degenerate vMF distributions.
This paper studies the Fisher-Rao geometry on the parameter space of beta distributions. We derive the geodesic equations and the sectional curvature, and prove that it is negative. This leads to uniqueness for the Riemannian centroid in that space. We use this Riemannian structure to study canonical moments, an intrin…
New method improves counterfactual distribution learning for high-dimensional outcomes.
problem Counterfactual distribution learning for high-dimensional outcomes with concentrated structure.
method Geometry-adaptive diffusion-guided smoothing estimators combining causal nuisance adjustment and local outcome geometry.
result Geometry-adaptive methods show steeper error decay in semi-synthetic experiments.
Non-regular -graded geometries II: classifying geometries, and generic six-in-nine distributionsmath.DG
Complementing the previous paper in the series, this paper classifies -graded parabolic geometries, listing their important properties: the group , the graded tangent bundle and its algebraïc bracket, the relevant cohomology spaces and the standard Tractor bundle $\mc{T}$. Several of these geometries …
Introduces Carrollian Lie algebroids to handle singular Carrollian geometries.
problem Handling singular Carrollian geometries within standard Carrollian geometry.
method Introduces Carrollian Lie algebroids to study singular Carrollian geometries.
result Established the existence of compatible connections on Carrollian Lie algebroids.
We analyze the classic problem of existence of Einstein metrics in a given conformal structure for the class of conformal structures inducedf Nurowski's construction by (oriented) (2,3,5) distributions. We characterize in two ways such conformal structures that admit an almost Einstein scale: First, they are precisely …
New method uses Fisher-Rao metric for non-Gaussian decoders.
problem Existing latent space geometry theory only works for Gaussian decoders.
method Pull back Fisher-Rao metric to latent space for non-Gaussian decoders.
result Achieves meaningful latent geometries for various non-Gaussian decoders.
Abstract reviews distributions and subbundles in differential geometry.
problem Understanding distributions and subbundles in differential geometry.
method Systematic review of distributions and subbundles, including sheaves and differentiability cases.
result Detailed consideration of Orbit Theorem and its applications.
Almost Robinson geometriesmath.DG
Study on almost Robinson geometries, focusing on their intrinsic torsion and leaf space properties.
problem Investigating the geometry of almost Robinson manifolds, a Lorentzian analog of almost Hermitian manifolds.
method Classification based on intrinsic torsion, analysis of leaf space properties, and conformal invariants.
result Comprehensive classification of almost Robinson manifolds based on their intrinsic torsion.
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 -parallel prior.
result The beta-logistic distribution admits an -parallel prior for any real number .
This study analyzes satellite communication latency using a stochastic geometry model.
problem Latency analysis of LEO satellite relay communication systems.
method Stochastic geometry framework with spherical BPP models, suboptimal satellite relay selection strategy.
result Derives distance distributions and analytical expressions for transmission delays.
We exploit the spinor description of four-dimensional Walker geometry, and conformal rescalings of such, to describe the local geometry of four-dimensional neutral geometries with algebraically degenerate self-dual Weyl curvature and an integrable distribution of alpha-planes (algebraically special real alpha-geometry)…
Transformers infer tasks from context via two modes, geometrically shaped task vectors explain their behavior.
problem Understanding how transformers infer tasks from context and the geometric properties of task vectors.
method Synthetic setting to train small transformers, mathematical characterization of task-vector geometry and inference modes.
result Task-vector geometry shapes in-distribution and out-of-distribution behavior of transformers.
Study compares geometric approaches for shape and deformation statistics.
problem Characterizing statistical models of shapes and deformations.
method Information geometry and Wasserstein geometry.
result Wasserstein estimator is robust against waveform perturbation.
Paper explores Elliptical Wishart distributions in signal processing and machine learning.
problem Estimating parameters of Elliptical Wishart distributions.
method Proposes fixed point and Riemannian optimization algorithms for maximum likelihood estimation.
result Characterizes existence, uniqueness, and convergence of the MLE.
Igeood detects out-of-distribution samples using information geometry.
problem Out-of-distribution (OOD) detection in machine learning systems.
method Igeood uses the Fisher-Rao geodesic distance to detect OOD samples from any pre-trained neural network.
result Igeood outperforms state-of-the-art methods on various network architectures and datasets.
Some of the well known Fefferman like constructions of parabolic geometries end up with a new structure on the same manifold. In this paper, we classify all such cases with the help of the classical Onishchik's lists \cite{onish1} and we treat in detail the only new series of inclusions providing the spinorial structur…
Tangency sets of non-involutive distributions and unrectifiability in Carnot-Carathéodory spacesmath.DG
New insights into integrability and rectifiability in sub-Riemannian geometry.
problem Understanding rectifiability in sub-Riemannian spaces.
method Refined Frobenius Theorem for non-involutive distributions, new metric space class.
result Carnot-Carathéodory spaces are extremal in rectifiability.
Jordan algebras in information geometry linked to metrics on probability distributions.
problem Understanding Jordan algebras in information geometry.
method Inspired by Kirillov's coadjoint orbits, a pseudo-Riemannian metric is constructed on Jordan algebra leaves.
result Not all points in the dual space lie on a leaf, and the metric structure depends on the cone of positive functionals.
Partial AHS-structures extend G-structures and Cartan geometries to manifolds with involutive distributions.
problem Extending G-structures and Cartan geometries to manifolds with involutive distributions.
method Developing a canonical Cartan geometry for partial AHS-structures and constructing BGG sequences.
result Partial AHS-structures have analogs of BGG sequences, providing fine resolutions of sheaves.
The paper explores how AI systems use information geometry to encode semantic structure.
problem How AI systems encode semantic structure into geometric representation spaces.
method Focuses on softmax distributions and develops dual steering method for robust concept manipulation.
result Dual steering optimally modifies target concepts while minimizing off-target changes.
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.
cCorrGAN approximates conditional correlation matrices using GANs.
problem Learning empirical conditional distributions in the elliptope of correlation matrices.
method Conditional Generative Adversarial Networks (GANs) applied to correlation matrices.
result Validated through Monte Carlo simulations in finance.
Classifies multiply-transitive (2,3,5)-distributions using modern Cartan geometry.
problem Classifying multiply-transitive -distributions.
method Modern Cartan-geometric approach, incorporating structure theory.
result Complete classifications in both complex and real settings, with full curvature and infinitesimal holonomy.
Develops a Riemannian archetypal analysis for interpretable non-linear data.
problem Limited performance of classical archetypal analysis on non-linear data.
method Riemannian geometry for data-driven pullback, geodesic convex combinations, convex relaxation followed by non-convex refinement.
result Combines interpretability of classical archetypal analysis with expressive power of modern non-linear models.
Distributions of Monge type are a class of strongly regular bracket-generating distributions introduced by I. Anderson, Zh. Nie and P. Nurowski. Their symbol algebras prolong to simple graded Lie algebras, thus allowing one to associate a parabolic geometry to any given Monge distribution. This article is devoted to th…
We show that any dimension nearly Kähler (or nearly para-Kähler) geometry arises as a projective manifold equipped with a holonomy reduction. In the converse direction we show that if a projective manifold is equipped with a parallel -dimensional cross product on its standard tractor bundle …
Online distributional prediction with latent cluster geometry
problem Predicting the full data-generating distribution in non-stationary streams
method Representing candidate laws as latent cluster geometry and using Gibbs quasi-posterior
result Achieving sublinear cumulative Wasserstein regret under bounded support and stable latent geometry