New flow generates surfaces with constant curvature.
problem Creating surfaces with specific curvature properties.
method Framed curvature flow, analyzing trajectory surfaces.
result Trajectory surfaces of constant mean or Gaussian curvature.
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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.
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48 results for “Trajectory Curvature”
Efficiently trains forward processes to minimize generative trajectories curvature.
problem High curvature of generative trajectories slows down sampling speed.
method Trains forward process to minimize curvature without ODE/SDE simulation.
result Lower curvature than previous models, decreased sampling costs.
Study of closed trajectories in hyperbolic plane with specific curvature constraints.
problem Critical trajectories in hyperbolic plane for a specific energy function.
method Classification of critical trajectories based on momentum causal character, proof of existence of closed trajectories.
result Existence of countably many closed trajectories with time-like momentum.
The paper defines and analyzes conformal trajectories in 3D space forms.
problem Understanding trajectories in curved 3D spaces.
method Defined conformal trajectories and studied their properties in , , and .
result Conformal trajectories in and have constant curvature and torsion.
The paper provides uniform length estimates for trajectories on flat cone surfaces.
problem Estimating the length of trajectories on flat cone surfaces.
method Using self-intersection numbers and constants depending only on the flat metric, the paper focuses on convex flat cone spheres with a positive curvature gap and a fixed number of singularities.
result Uniform two-sided estimates for trajectory lengths on convex flat cone spheres are obtained.
We study the existence of closed trajectories of a particle model on null curves in anti-de Sitter 3-space defined by a functional which is linear in the curvature of the particle path. Explicit expressions for the trajectories are found and the existence of infinitely many closed trajectories is proved.
The paper characterizes Legendre curves on trans-S-manifolds.
problem Characterizing Legendre trajectories on trans-S-manifolds.
method Obtained curvature characterizations and classified Legendre curves.
result Classified Legendre curves with linearly dependent Frenet frame fields.
Billiard trajectories in curved spaces have predictable travel times.
problem Understanding travel times in billiard trajectories on curved surfaces.
method Analyzing geodesic flows and sectional curvature to prove time-preserving conjugacy.
result Billiard trajectories with almost identical obstacles have identical shapes.
With a view to constructing a Morse/Floer homology theory for CMC hypersurfaces, we prove a compactness result modulo broken trajectories for eternal mean curvature flows with forcing term in compact, hyperbolic manifolds.
It is shown that the equation which describes constant mean curvature surface via the generalized Weierstrass-Enneper inducing has Hamiltonian form. Its simplest finite-dimensional reduction has two degrees of freedom, integrable and its trajectories correspond to well-known Delaunay and do Carmo-Dajzcer surfaces (i.e.…
Hamilton flows on Kähler manifold for which all trajectories are -planar curves (complex analog of geodesics) are considered. These flows are called -planar. The equation which has to obey the Hamiltonian of -planar Hamilton flow is received and the method of finding general solution of this equation is propos…
Reasoning models generate differently based on problem difficulty, not just length.
problem Understanding how reasoning models handle different problem difficulties.
method Examined hidden-state trajectories across competitive programming, mathematics, and Boolean satisfiability.
result Corrected trajectory geometry shows difficulty-dependent differences in reasoning models, with stronger effects in the code domain.
Absolute parallelism geometry is frequently used for physical applications. It has two main defects, from the point of view of applications. The first is the identical vanishing of its curvature tensor. The second is that its autoparallel paths do not represent physical trajectories. The present work shows how these de…
Minimal hypersurfaces can't always be connected by mean curvature flow.
problem Existence of connecting mean curvature flows for minimal hypersurfaces.
method Minimal hypersurface analogue of gradient flow trajectories between critical points.
result Additional topological and variational obstructions to connecting mean curvature flows.
The paper investigates how neural network weights evolve to monitor training progress.
problem Monitoring the training progress of neural networks in a cost-effective manner.
method Investigates the evolution of neural network weights in weight space.
result DNN models evolve on unique, smooth trajectories in weight space that can be used to track training progress.
The paper introduces a new method to assess system stability using torsion of state trajectories.
problem Stability assessment of linear time-invariant systems.
method Using the torsion of the state trajectory to determine stability.
result Conditions for stability and asymptotic stability are established based on the behavior of torsion.
We describe the curves of constant (geodesic) curvature and torsion in the three-dimensional round sphere. These curves are the trajectory of a point whose motion is the superposition of two circular motions in orthogonal planes. The global behavior may be periodic or the curve may be dense in a Clifford torus embedded…
A canal surface is the envelope of a moving sphere with varying radius, defined by the trajectory C(t) (spine curve) of its center and a radius function r(t). In this paper, we investigate when parameter curves of the canal surface are also lines of curvature. Last of all, for special spine curves we obtain the radius …
Transformer models are compared to curved spacetime in General Relativity.
problem Understanding the geometric structure of Transformer models.
method Geometric analogy to General Relativity, focusing on attention and curvature effects.
result Transformer token embeddings exhibit curvature, affecting their evolution over layers.
The paper studies magnetic curves in -manifolds and their properties.
problem Understanding magnetic trajectories in -manifolds.
method Proving magnetic trajectories are -slant curves and providing parametrizations.
result Normal magnetic curves in -manifolds are -slant curves with specific curvature functions.
The paper examines the sampling dynamics of diffusion models using ODEs.
problem Understanding the sampling dynamics of diffusion models.
method Careful inspection of ODE-based sampling of SDEs, revealing structures and relationships.
result Established a theoretical relationship between optimal ODE-based sampling and mean-shift algorithm.
The local motion of a null curve in Minkowski 3-space induces an evolution equation for its Lorentz invariant curvature. Special motions are constructed whose induced evolution equations are the members of the KdV hierarchy. The null curves which move under the KdV flow without changing shape are proven to be the traje…
SGD without replacement decouples into curvature-following and flatness-regularizing steps.
problem Theoretical analysis of SGD without replacement for large-scale neural networks.
method Analysis of SGD without replacement in a realistic regime, considering high curvature and flatness.
result Optimizing with SGD without replacement is locally equivalent to an additional regularizer step.
Gradient descent training of neural networks leads to solutions close to natural cubic splines.
problem Understanding the implicit bias of gradient descent in neural networks.
method Analysis of gradient descent training for wide neural networks, focusing on the curvature penalty and initialization schemes.
result The solutions of gradient descent training are polyharmonic splines for certain initialization schemes.
ATLAS adapts HMC step size and trajectory length for complex geometries.
problem Sampling complex geometries with constant step size HMC/NUTS.
method Adapts step size and trajectory length using local Hessian and no U-turn condition.
result ATLAS accurately samples complex geometries, outperforming NUTS.
GH-PID uses guided harmonic paths for efficient SOT with interpretable diagnostics.
problem Efficiently solving Stochastic Optimal Transport with hard terminal distributions and soft costs.
method Guided Harmonic Path-Integral Diffusion (GH-PID) framework with low-dimensional guidance.
result GH-PID generates geometry-aware, cost-reducing trajectories that match terminal distributions.
Improved tracking of tangled point sources using Riemannian metrics.
problem Tangled point source trajectories in temporal stacks.
method Lifting to higher-dimensional space of roto-translation group, new regularisation based on relaxed Reeds-Shepp metric.
result Reconstruction and untangling of trajectories even from numerical standpoint.
A -dimensional Riemannian manifold is called Killing submersion if it admits a Riemannian submersion over a surface such that its fibers are the trajectories of a complete unit Killing vector field. In this paper, we give a characterization of proper biharmonic CMC surfaces in a Killing submersion. In the last part,…
We study the problem of discriminative sub-trajectory mining. Given two groups of trajectories, the goal of this problem is to extract moving patterns in the form of sub-trajectories which are more similar to sub-trajectories of one group and less similar to those of the other. We propose a new method called Statistica…
Paper uses deep imitation learning to predict aircraft trajectories accurately.
problem Inefficient and costly Air Traffic Management system limits predictability.
method Generative Adversarial Imitation Learning framework with trajectory clustering and classification.
result Accurate predictions for entire trajectory stages, pre- and tactical.
Paper infers human mobility from sparse trajectories.
problem Modeling and inferring human mobility from sparse trajectory data.
method Proposes a single trajectory inference algorithm and a deep learning architecture for multiple trajectories.
result Deep learning model achieves 2x overall accuracy improvement on sparse trajectories.
Pattern Ensembling for Spatial Trajectory Reconstructionphysics.data-an
Pattern ensembling fills in missing or inaccurate trajectory data.
problem Incompleteness, missing information, and inaccuracies in geolocation data.
method Probabilistically ensemble similar trajectory patterns from the vicinity.
result Reconstructs missing or unreliable trajectory segments effectively.
Study shows magnetic trajectories in Berger spheres are homogeneous.
problem Homogeneity of contact magnetic trajectories in Berger spheres.
method Proved every contact magnetic trajectory is a product of a homogeneous geodesic and a charged Reeb flow.
result Contact magnetic trajectories in Berger spheres are homogeneous.
Improved trajectory prediction for team sports using sparse outputs.
problem Challenging to train deep learning models for player trajectory prediction.
method Sparse trajectory prediction and constant acceleration interpolation.
result Interpolation improves performance for all tested models.
WS-II algorithm segments trajectories with high accuracy.
problem Trajectory segmentation for diverse data sources.
method Supervised learning with sliding window analysis.
result WS-II outperforms other methods in all datasets.
A framework clusters vehicle motion trajectories efficiently.
problem Costly manual annotation of vehicle motion data.
method Five-stage framework: align, embed, extract, embed, cluster.
result Framework achieves promising results on real-world dataset.
Proposes a CNN-based method for better trajectory owner prediction.
problem Improves trajectory owner prediction for better personalized recommendations and urban planning.
method Connects POIs in a graph, encodes POIs into vectors, transforms trajectories into matrices, and uses a CNN to detect features and predict owners.
result Significantly outperforms existing methods in various metrics.
In this paper we propose a new parameter-free method for trajectory classification which finds the best trajectory partition and dimension combination for robust trajectory classification. Preliminary experiments show that our approach is very promising.
TrajGAIL: Generating Urban Vehicle Trajectories using Generative Adversarial Imitation Learningcs.LG
Generative model learns vehicle trajectory distributions for better data generalization.
problem Data sparsity and privacy issues in urban vehicle trajectory analysis.
method Generative adversarial imitation learning framework for urban vehicle trajectory generation.
result TrajGAIL model produces synthetic trajectories similar to real ones, achieving significant performance gains.
Many AI problems, in robotics and other domains, are goal-directed, essentially seeking a trajectory leading to some goal state. In such problems, the way we choose to represent a trajectory underlies algorithms for trajectory prediction and optimization. Interestingly, most all prior work in imitation and reinforcemen…
We give a complete answer to the question of when two curves in two different Riemannian manifolds can be seen as trajectories of rolling one manifold on the other without twisting or slipping. We show that up to technical hypotheses, a rolling along these curves exists if and only if the geodesic curvatures of each cu…
Method detects trajectory outliers using Hodge Laplacian embeddings.
problem Detecting outliers in trajectory data on simplicial complexes.
method Flow-embeddings using Hodge 1-Laplacian of simplicial complexes.
result Classifies trajectories based on topological behavior.
We generalize Hadamard-Stoker-Currier Theorems for surfaces immersed in a Killing submersion over a strictly Hadamard surface whose fibers are the trajectories of a unit Killing field. We prove that every complete surface whose principal curvatures are greater than a certain function (depending on the ambient manifold)…
The paper finds the Finsler structure of Apollonian weak metric on unit disc.
problem Understanding the Finsler structure of Apollonian weak metric on the unit disc.
method Analyzing the deformation of hyperbolic Poincaré metric by a closed 1-form.
result The Apollonian weak-Finsler structure has bounded below -curvature and flag curvature satisfying .
We consider the optimal control problem for null curves in de Sitter 3-space defined by a functional which is linear in the curvature of the trajectory. We show how techniques based on the method of moving frames and exterior differential systems, coupled with the reduction procedure for systems with a Lie group of sym…
Study on billiard trajectories with fixed bounces.
problem Counting periodic trajectories with specific bounces.
method Analyzes two-dimensional dispersive billiard systems.
result Asymptotic growth of primitive periodic trajectories.
Representation learning of pedestrian trajectories transforms variable-length timestamp-coordinate tuples of a trajectory into a fixed-length vector representation that summarizes spatiotemporal characteristics. It is a crucial technique to connect feature-based data mining with trajectory data. Trajectory representati…
GeomHerd predicts herding behavior before market prices move, using Ricci curvature of agent interaction graphs.
problem Quantifying herding behavior in markets that lags behind actual price movements.
method Develops a geometric framework to track coordination on agent interaction graphs, bypassing lag in price-correlation statistics.
result GeomHerd anticipates herding long before market baselines, with significant lead times in predictions.