Geodesics found in deep linear networks.
problem Finding shortest paths in deep neural networks.
method Derived ODEs and explicit solutions for geodesics.
result Horizontal straight lines are geodesics in invariant manifold.
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arXiv research
A locally-built, LLM-digested index of recent arXiv papers in quant finance, geometry/topology, and statistical ML — keyword search served straight from SQLite on this machine.
168,742 papers · 148 categories
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48 results for “geodesic networks”
Analyzes geodesic lengths in sparse networks, deriving a distribution.
problem Understanding connectivity and robustness in networked systems.
method Analytic derivation of geodesic length distribution in sparse networks.
result Simple closed-form expression for geodesic length distribution.
Geodesics connect model modes in neural network loss landscapes.
problem Connecting modes in neural network loss landscapes.
method Reframed mode connectivity in Information Geometry, hypothesized geodesics as mode-connecting paths, proposed algorithm to approximate geodesics.
result Geodesics achieve mode connectivity in neural networks.
Study finds geodesic networks for surfaces with convex boundary.
problem Finding geodesic networks for surfaces with convex boundary.
method Investigates free boundary geodesic networks in surfaces with non-negative sectional curvature and convex boundary.
result Existence of a geodesic network realizing the first width of a surface with non-negative sectional curvature and strictly convex boundary.
Theorem shows generic metrics yield non-degenerate geodesic nets.
problem Characterizing geodesic nets on generic metrics.
method Proving all connected embedded nets are non-degenerate for Baire-generic metrics.
result All stationary geodesic nets are non-degenerate for generic metrics.
The paper proposes a method for generating uniform interpolations on data manifolds.
problem Generating high-quality interpolations between data samples on complex manifolds.
method Autoencoder network with interpolation network, regularized by a Riemannian metric.
result The method generates interpolations that remain within the manifold's distribution.
ResNets learn the geodesic curve in Wasserstein space.
problem Characterize the dynamics of deep residual networks during training.
method Modeling ResNet dynamics using continuity equations and optimal transport.
result ResNets learn the geodesic curve in the Wasserstein space.
Paper analyzes convergence of proximal algorithm in metric spaces without geodesic convexity.
problem Analyzing convergence of proximal algorithm in general metric spaces.
method Analysis of the Wasserstein proximal algorithm without geodesic convexity assumption.
result Establishes unbiased and linear convergence rate for proximal algorithm under natural Wasserstein inequality.
This paper develops GPCA for probability distributions using Otto-Wasserstein geometry.
problem Analyzing modes of variation in datasets of probability measures.
method Geodesic Principal Component Analysis (GPCA) on Wasserstein space with neural networks.
result Identification of geodesic curves that capture modes of variation in probability distributions.
The paper sparsifies networks by finding efficient paths in their functional space.
problem Sparsifying neural networks to improve performance and efficiency.
method The authors use the geometry of weight spaces and functional manifolds to find efficient paths (geodesics) in the functional space of neural networks.
result The proposed framework can sparsify networks and improve performance on various tasks.
Proposes Geodesic Integrated Gradients (GIG) for more accurate feature attributions in deep networks.
problem Flawed attributions using straight paths from Integrated Gradients (IG).
method Introduces a model-induced Riemannian metric and computes attributions along geodesics.
result GIG produces more faithful attributions than IG on benchmarks.
Neural solver computes Wasserstein geodesics and velocity fields efficiently.
problem Computing Wasserstein geodesics and velocity fields efficiently.
method Sample-based neural network approach to solve the minimax problem.
result Directly samples from target distribution and estimates velocity field.
fMRI is a unique non-invasive approach for understanding the functional organization of the human brain, and task-based fMRI promotes identification of functionally relevant brain regions associated with a given task. Here, we use fMRI (using the Poffenberger Paradigm) data collected in mono- and dizygotic twin pairs t…
Fréchet Geodesic Boostingstat.ML
FGBoost boosts gradient boosting for complex data.
problem Gradient boosting struggles with non-Euclidean data.
method Introduces FGBoost for geodesic metric spaces.
result FGBoost performs well on complex data.
Improves latent space structure for better data representation.
problem Limited ability of conventional priors to encode data manifold structure.
method Introduces an Encoded Prior Sliced Wasserstein AutoEncoder with iterative training and geodesic interpolation.
result Learned manifold encoding preserves topological and geometric properties of data.
Study on geodesic distances on SE(3)/SO(2) in machine learning.
problem Investigating the efficiency of computationally efficient sections in selecting geodesic distances.
method Analyzing geodesic distances on reductive homogeneous spaces, proving the efficiency of minimal distance sections.
result Minimal distance sections are not always geodesic minimizers, but minimal horizontal geodesics are.
New insights into how neural networks learn features, especially when they are very wide.
problem Understanding how gradient flow in wide neural networks selects solutions, especially in the feature-learning regime.
method Axiomatizing the canonical regularizer as a function-space energy and lift, and deriving geodesic ridge for the feature-learning regime.
result Gradient flow in feature-learning networks biases towards ridge regularization, distorting the inductive bias and damaging pretrained networks.
We prove that on a closed surface, for any , our min-max theory for prescribing mean curvature produces a solution given by a curve of constant geodesic curvature which is almost embedded, except for finitely many points, at which the solution is a stationary junction with integer density. Moreover, each smoot…
GeONet learns the Wasserstein geodesic without mesh discretization.
problem Computing the Wasserstein geodesic between complex data distributions.
method Mesh-invariant deep neural operator network that learns saddle point optimality conditions.
result GeONet achieves comparable accuracy to standard OT solvers with reduced computational cost.
Generative models learn manifold structure; new approach uses atlas and geodesic interpolation.
problem Challenges in representing manifolds with topology different from Euclidean space.
method Atlas Generative Models (AGMs) with hybrid latent spaces and geodesic interpolation.
result Geodesic interpolation can be extended to AGMs, improving manifold representation.
We consider the problem of sampling from posterior distributions for Bayesian models where some parameters are restricted to be orthogonal matrices. Such matrices are sometimes used in neural networks models for reasons of regularization and stabilization of training procedures, and also can parameterize matrices of bo…
Minimal networks on S^2math.DG
The paper extends minimal network theory to the sphere, proving local minimality.
problem Finding networks of minimal length on the sphere.
method Adapted spherical geometry, calibration method, and local metric perturbation estimates.
result Spherical minimal networks composed of great-circle arcs are locally length-minimizing within small geodesic balls.
A neural network learns efficient parametrizations of product shape spaces.
problem Efficiently parametrize complex shape spaces with high computational costs.
method Developed a neural network architecture that separately learns approximations for low-dimensional factors and combines them.
result Demonstrated the effectiveness of the approach on synthetic and real data.
We introduce elastic geodesic grids for easy-to-fabricate, deployable structures.
problem Approximating freeform surfaces with deployable structures.
method Geodesic curves on target surfaces, kinematic mechanism, differential geometry.
result Elastic geodesic grids can approximate freeform surfaces easily and deployably.
Existing dimensionality reduction methods are adept at revealing hidden underlying manifolds arising from high-dimensional data and thereby producing a low-dimensional representation. However, the smoothness of the manifolds produced by classic techniques over sparse and noisy data is not guaranteed. In fact, the embed…
In this paper, we propose a novel deep learning framework for anatomy segmentation and automatic landmark- ing. Specifically, we focus on the challenging problem of mandible segmentation from cone-beam computed tomography (CBCT) scans and identification of 9 anatomical landmarks of the mandible on the geodesic space. T…
A hyperbolic space has been shown to be more capable of modeling complex networks than a Euclidean space. This paper proposes an explicit update rule along geodesics in a hyperbolic space. The convergence of our algorithm is theoretically guaranteed, and the convergence rate is better than the conventional Euclidean gr…
A half-geodesic is a closed geodesic realizing the distance between any pair of its points. All geodesics in a round sphere are half-geodesics. Conversely, this note establishes that Riemannian spheres with all geodesics closed and sufficiently many half-geodesics are round.
Study on homogeneous geodesics in sub-Riemannian geometry.
problem Characterizing and understanding homogeneous geodesics in sub-Riemannian manifolds.
method Criterion for geodesics to be homogeneous, proof of geodesic orbit spaces, examples of geodesic orbit sub-Riemannian manifolds.
result Existence of at least one homogeneous geodesic under broad conditions.
In non-compact manifolds, geodesic flowers exist.
problem Existence of geodesic flowers in non-compact manifolds.
method Proving the existence of non-trivial geodesic flowers in complete non-compact manifolds with locally convex ends.
result Non-trivial geodesic flowers exist in every complete non-compact manifold with locally convex ends.
Study on Mabuchi functional's convexity using ε-geodesics.
problem Understanding the convexity of the Mabuchi functional.
method Analysis of ε-geodesics to study the Mabuchi functional's convexity.
result Uniform fiberwise non-degeneracy of geodesics when Mabuchi functional is ε-affine.
Geodesic graphs for special Finsler metrics on spheres are studied.
problem Characterizing geodesic orbit Finsler metrics on spheres.
method Explicit constructions and group extensions.
result Not all projective spaces admit invariant Finsler metrics.
Characterizes visibility and geodesic loops in complex domains.
problem Visibility and geodesic loops in complex domains.
method Using quasi-geodesic frames to characterize visibility and geodesic loops.
result Characterizes visibility and existence of geodesic loops in Kobayashi complete hyperbolic and Gromov hyperbolic domains.
Conformal geodesics can't spiral in Riemannian manifolds.
problem Existence of spiral conformal geodesics on Riemannian manifolds.
method Analyzing properties of conformal geodesics on Riemannian manifolds.
result No conformal geodesic can become trapped in every neighborhood of a point.
Geodesics and -geodesics on tangent bundle with -Sasaki metric over para-Kähler-Norden manifoldmath.DG
Study geodesics and F-geodesics on tangent bundles over para-Kähler-Norden manifolds.
problem Investigate geodesics and F-geodesics on tangent bundles.
method Investigate geodesics and F-geodesics on tangent bundles and φ-unit tangent bundles equipped with φ-Sasaki metric over para-Kähler-Norden manifolds.
result Investigate and analyze geodesics and F-geodesics on tangent bundles.
New quasi-geodesics for Stiefel manifold simplify complex computations.
problem Efficiently solving geodesic endpoint problem on Stiefel manifold.
method Derived new representations of quasi-geodesics for large-scale computations.
result New quasi-geodesics are closer to Riemannian geodesics.
Study on geodesics of Finsler metrics derived from Riemannian metrics.
problem Investigating geodesics in Finsler metrics derived from Riemannian metrics.
method Proved geodesic lemma for a family of Riemannian geodesic orbit metrics, analyzed geodesic graphs.
result Derived Finsler metrics have geodesic orbit property and belong to a new class of metrics.
Growth rates of geodesics on modular orbifolds are studied.
problem Understanding growth rates of geodesics on modular orbifolds.
method Exhaustion of modular orbifold by compact subsurfaces, analysis of low lying geodesics and reciprocal geodesics.
result Growth rates of low lying geodesics and reciprocal geodesics converge to the full set's growth rate.
We study the geometry of the Thurston metric on Teichmuller space by examining its geodesics and comparing them to Teichmuller geodesics. We show that, similar to a Teichmuller geodesic, the shadow of a Thurston geodesic to the curve graph is a reparametrized quasi-geodesic. However, we show that the set of short curve…
We describe the geometry of geodesics on a Lorentz ellipsoid: give explicit formulas for the first integrals (pseudo-confocal coordinates), curvature, geodesically equivalent Riemannian metric, the invariant area-forms on the time- and space-like geodesics and invariant 1-form on the space of null geodesics. We prove a…
This paper classifies geodesics of projectively flat sprays and introduces a method to determine sprays based on geodesics.
problem Classifying geodesics of projectively flat sprays and determining sprays based on geodesics.
method Introduction of a geodesic method to determine an n-dimensional spray based on a family of curves with 2(n-1) free parameters as geodesics.
result Classification of geodesics of projectively flat sprays and determination of sprays based on geodesics.
The paper connects geodesic flows, hyperbolic geodesics, and stable ergodicity.
problem Understanding conditions for geodesic flows to be Anosov and ergodic.
method Analyzing Finsler and Riemannian metrics on surfaces, using recent results.
result Geodesic flows on surfaces are stably ergodic if and only if they are Anosov.
Tight geodesics were introduced by Masur-Minsky in [17]. They and their hierarchies have been a powerful tool in the study of the curve complex, mapping class groups, Teichmüller spaces, and hyperbolic 3-manifolds. In the same paper, they showed that there are at least one and at most finitely many tight geodesics betw…
No closed timelike geodesics in Kerr spacetimes, proving absence of closed causal geodesics.
problem Proving the nonexistence of closed timelike geodesics in Kerr spacetimes.
method Analyzing the Kerr-star spacetime, excluding closed null geodesics and proving the nonexistence of closed timelike geodesics.
result No closed timelike geodesics in Kerr spacetimes.
Generic geodesic nets are dense in high-dimensional manifolds.
problem Density of non-closed geodesic nets in high-dimensional manifolds.
method Proving density for a generic metric on a manifold.
result Stationary geodesic nets that are not closed geodesics form a dense set.
Let M be a Margulis spacetime whose associated complete hyperbolic surface S has compact convex core. Generalizing the correspondence between closed geodesics on M and closed geodesics on S, we establish an orbit equivalence between recurrent spacelike geodesics on M and recurrent geodesics on S. In contrast, no timeli…
The authors define a class of functions on Riemannian manifolds, which is called geodesic semilocal E-preinvex functions, as a generalization of geodesic semilocal E-convex and geodesic semi E-preinvex functions and some of its properties are established. Furthermore, a nonlinear fractional multiobjective programming i…
New method calibrates deep models for both in-distribution and out-of-distribution samples.
problem Ensuring calibration for deep models in safety-critical applications, especially in OOD regions.
method Geodesic distance and Gaussian kernel to calibrate deep models.
result Proposed KDF and KDN methods achieve well-calibrated posteriors for both in-distribution and out-of-distribution samples.