arXiv research

Optimal tracking of nonholonomic systems using geometric methods.

Critical trajectories in a sphere are found for a specific bending functional.

Billiard trajectories and geodesics are closely related geometrically.

Study of 2D Lorentzian anti-de Sitter plane using geometric control theory.

Neuroscientific studies of drawing-like movements usually analyze neural representation of either geometric (eg. direction, shape) or temporal (eg. speed) features of trajectories rather than trajectory's representation as a whole. This work is about empirically supported mathematical ideas behind splitting and merging…

Diffusion models' sampling paths lie in a low-dimensional subspace, resembling boomerangs.

Study optimal paths in Zermelo's navigation problem using geometric equations.

Geometric Brownian motion (GBM) is a key model for representing self-reproducing entities. Self-reproduction may be considered the definition of life [5], and the dynamics it induces are of interest to those concerned with living systems from biology to economics. Trajectories of GBM are distributed according to the we…

This work enables UAVs to autonomously form desired trajectories without needing a central plan.

We use the methods of geometric control theory to study extremal trajectories of vertical rolling disk. We focus on the role of symmetries of the underlying geometric structures. We demonstrate the computations in the CAS Maple package DifferentialGeometry.

Enhances reinforcement learning from sparse data.

We study local control of the mechanism with the growth vector (4,7). We study controllability and extremal trajectories on the nilpotent approximation as an example of the control theory on Lie group. We give solutions of the system an show examples of local extremal trajectories.

This paper describes the construction of a canonical compactification of the space of trajectories and of the unstable/stable sets of a generic gradient like vector field on a closed manifold as well as a canonical structure of a smooth manifold with corners of these spaces. As an application we discuss the geometric c…

TRAKNN detects rare atmospheric trajectories efficiently.

The energy in a square membrane $Ω$ subject to constant viscous damping on a subset $ω\subset Ω$ decays exponentially in time as soon as $ω$ satisfies a geometrical condition known as the "Bardos-Lebeau-Rauch" condition. The rate $τ(ω)$ of this decay satisfies $τ(ω)= 2 \min(-μ(ω), g(ω))$ (see Lebeau [Math. Phys. Stud. …

New method learns dynamics from sparse data using geometric constraints.

Develops MENT for interpreting and detecting changes in network trajectories.

Study shows how steepest descent algorithms' geometric margin increases during training.

GDB bridges geometric states with improved accuracy and generality.

Geometrically interpolates rigid body motions with initial and terminal twists.

In this paper, we propose a geometric framework to analyze the convergence properties of gradient descent trajectories in the context of linear neural networks. We translate a well-known empirical observation of linear neural nets into a conjecture that we call the \emph{overfitting conjecture} which states that, for a…

We show an example providing a significance in geometric control theory of the existence of the dependence locus of a system of vector fields in particular, the generic appearance of non-trivial singular trajectories embedded in the dependence locus.

Stochastic gradient descent's long-term fluctuations are described by a diffusion limit.

This work characterizes how data augmentation shapes neural representations.

Unified framework for multi-view diffusion geometries using intertwined diffusion trajectories.

In this paper we introduce a link between geometry of ordinary continued fractions and trajectories of points that moves according to the second Kepler law. We expand geometric interpretation of ordinary continued fractions to the case of continued fractions with arbitrary elements.

Geometric Brownian motion (GBM) is a model for systems as varied as financial instruments and populations. The statistical properties of GBM are complicated by non-ergodicity, which can lead to ensemble averages exhibiting exponential growth while any individual trajectory collapses according to its time-average. A com…

Paper analyzes EM algorithm's trajectory in 2MLR, revealing cycloid behavior.

This paper details a series of experiments in searching for minimal energy configurations for knots and links using the computer program KnotPlot. The most interesting phenomena found in these experiments is the dependence of the trajectories of energy descent upon the initial geometric conditions of the knotted embedd…

In this paper we derive the optimal execution trajectory for a trader who wishes to buy or sell a large position of shares which evolve as a geometric Brownian process in contrast to the arithmetic model which prevails in the existing literature, and with a general temporary impact $h$. We provide a couple of examples …

New method constructs geometric flat outputs for robotic systems using symmetry.

The paper proves a nonholonomic version of Maupertuis-Jacobi principle and shows that nonholonomic trajectories minimize length.

Discrete Flow Maps bypass sequential prediction limits for parallel text generation.

New model uses minimal data to outperform traditional hedging strategies.

The paper examines the sampling dynamics of diffusion models using ODEs.

The purpose of this paper is to use the framework of Lie algebroids to study optimal control problems for affine connection control systems on Lie groups. In this context, the equations for critical trajectories of the problem are geometrically characterized as a Hamiltonian vector field.

A new geometric metric identifies true data changes from parametrization artifacts in high-dimensional representations.

Geometrically characterizes virtual nonlinear nonholonomic constraints using symplectic methods.

The paper tracks patient recovery using graphs of joint movement data.

The paper defines symmetries in no-arbitrage markets.

Signature tensors uniquely identify ODE solutions.

Stochastic gradient descent (SGD) is a key ingredient in the training of deep neural networks and yet its geometrical significance appears elusive. We study a deterministic model in which the trajectories of our dynamical systems are described via geodesics of a family of metrics arising from the diffusion matrix. Thes…

LLMs learn probability density functions in-context, showing distinct learning trajectories.

We geometrically describe optimal control problems in terms of Morse families in the Hamiltonian framework. These geometric structures allow us to recover the classical first order necessary conditions for optimality and the starting point to run an integrability algorithm. Moreover the integrability algorithm is adapt…

The paper introduces metrics for robust unsupervised learning of vehicle interactions.

The paper is devoted to modeling optimal exercise strategies of the behavior of investors and issuers working with convertible bonds. This implies solution of the problems of stock price modeling, payoff computation and min-max optimization. Stock prices (underlying asset) were modeled under the assumption of the geome…

Langmuir-Blodgett films (LB-films) consist from few LB-monolayers which are high structured nanomaterials that are very promising materials for applications. We use a geometrical approach to describe structurization into LB-monolayers. Consequently, we develop on the 1-jet space J^1([0,\infty),R^2) the single-time Lagr…

Symmetry groups of PDEs allow to transform solutions continuously into other solutions. In this paper, we use this property for the observability analysis of nonlinear PDEs with input and output. Based on a differential-geometric representation of the nonlinear system, we derive conditions for the existence of special …