Optimal tracking of nonholonomic systems using geometric methods.
problem Tracking a trajectory for nonholonomic mechanical systems.
method Geometric optimal control, Pontryagin Maximum Principle, variational approach.
result Optimal control solutions for nonholonomic systems validated by examples and simulations.
Critical trajectories in a sphere are found for a specific bending functional.
problem Finding closed trajectories in a sphere for a specific bending functional.
method Existence of infinitely many closed trajectories shown for a given Lagrange multiplier.
result Existence of closed trajectories dependent on a pair of relatively prime natural numbers.
Billiard trajectories and geodesics are closely related geometrically.
problem Understanding the relationship between billiard trajectories and geodesics on surfaces.
method Establishing mutual approximation results for billiard trajectories and geodesic segments on surfaces.
result For Riemannian billiard tables, there are families of fold-type surfaces such that every sequence of geodesic segments on these surfaces has a subsequence that converges to a billiard trajectory.
Study of 2D Lorentzian anti-de Sitter plane using geometric control theory.
problem Understanding extremal trajectories and reachable set on anti-de Sitter plane.
method Geometric control theory and differential geometry.
result Construction of optimal synthesis and description of Lorentzian distance.
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.
problem Understanding the geometric structure of diffusion-based generative models.
method Characterization of deterministic sampling trajectories using low-dimensional subspace and kernel-estimated data modeling.
result Sampling trajectories in diffusion models are confined to a low-dimensional subspace and exhibit a boomerang shape.
Study optimal paths in Zermelo's navigation problem using geometric equations.
problem Optimal control paths in Zermelo's navigation problem.
method Geometric and differential equations approach to obtain precise ODE system.
result Obtained precise equations for optimal trajectories.
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.
problem Autonomous formation of complex trajectories in UAVs.
method Decentralized control system using geometric embeddings.
result Quadcopters self-organize into desired trajectories while maintaining separation.
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.
problem Limited data for offline reinforcement learning.
method Trajectory-based data augmentation.
result Improves reinforcement learning performance.
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.
problem Detecting rare atmospheric anomalies over long periods.
method Unsupervised, recurrence-based kNN algorithm.
result Rare trajectories correspond to physical anomalies.
The energy in a square membrane Ω subject to constant viscous damping on a subset ω⊂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)=2min(−μ(ω),g(ω)) (see Lebeau [Math. Phys. Stud. …
New method learns dynamics from sparse data using geometric constraints.
problem Learning dynamics from sparse, undersampled data.
method Reformulates inference as a stochastic control problem, using geometry-driven path augmentation.
result Accurately recovers stochastic dynamics from extremely undersampled data.
Develops MENT for interpreting and detecting changes in network trajectories.
problem Distortion of network geometry and invalidation of temporal comparisons in dynamic network analysis.
method Develops Multiscale Euclidean Network Trajectories (MENT) framework based on second-moment geometry.
result Validates and interprets network trajectories through isotropic normalization and orthogonal transformations.
Study shows how steepest descent algorithms' geometric margin increases during training.
problem Understanding implicit bias in steepest descent algorithms for neural networks.
method Analysis of steepest descent algorithms with infinitesimal learning rates in homogeneous neural networks.
result Limit points of training trajectories correspond to KKT points of margin-maximization problems.
GDB bridges geometric states with improved accuracy and generality.
problem Challenges in predicting geometric state evolution in complex systems.
method Geometric Diffusion Bridge (GDB) framework using equivariant diffusion bridges.
result GDB surpasses existing methods in accurately bridging geometric states.
Geometrically interpolates rigid body motions with initial and terminal twists.
problem Finding spatial trajectories between prescribed initial and terminal poses.
method Derives solutions for k-IV-TIP and k-BV-TIP for k=1,...,4.
result Automatic cubic interpolation identical to minimum acceleration curve when twists are zero.
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.
problem Long-term behavior of stochastic gradient descent in non-smooth settings.
method Functional central limit theorem applied to rescaled trajectory of SGD.
result Characterization of long-term fluctuations around the minimizer.
This work characterizes how data augmentation shapes neural representations.
problem Understanding the impact of data augmentation on neural network representations.
method Embedding neural network hidden representations into a metric space invariant to transformations, analyzing shape-space trajectories.
result Increasing data augmentation strength leads to well-behaved trajectories in the embedded space, and different augmentation types steer representations in distinct directions.
Unified framework for multi-view diffusion geometries using intertwined diffusion trajectories.
problem Constructing multi-view diffusion geometries with flexible view interaction and fusion.
method Intertwined multi-view diffusion trajectories (MDTs) as a class of inhomogeneous diffusion processes.
result Established theoretical properties and derived diffusion distances and embeddings.
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.
problem Understanding the convergence and trajectory of EM algorithm in 2MLR.
method Explicit closed-form expressions for EM updates, recurrence relation derivation at population level.
result EM iterations lie on a cycloid trajectory, leading to theoretical estimate of convergence exponent.
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.
problem Finding flat outputs for arbitrary robotic systems remains an open question.
method Employing symmetry directly to construct a flat output.
result Demonstrated geometric flat outputs for various robotic systems.
The paper proves a nonholonomic version of Maupertuis-Jacobi principle and shows that nonholonomic trajectories minimize length.
problem Nonholonomic dynamics and their length minimization.
method Contact bundle formulation and geometric equivalence between problems.
result Regular solutions of nonholonomic mechanical problems are reparametrizations of geodesics with minimized Riemannian length.
Discrete Flow Maps bypass sequential prediction limits for parallel text generation.
problem Sequential autoregressive prediction limits large language model speed.
method Flow Maps compress generative trajectories into single-step mappings.
result Discrete Flow Maps surpass previous state-of-the-art results in discrete flow modeling.
New model uses minimal data to outperform traditional hedging strategies.
problem Optimizing hedging strategies with transaction costs and limited data.
method Model-free deep learning approach using a small number of trajectories.
result Neural network outperforms Black & Scholes and Leland models.
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 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.
Researchers interpret SGD using diffusion metrics for clearer geometric understanding.
problem Elusiveness of geometrical significance in stochastic gradient descent.
method Study a deterministic model with geodesics of diffusion metrics.
result Establishes parallel with General Relativity models.
A new geometric metric identifies true data changes from parametrization artifacts in high-dimensional representations.
problem Quantifying representation drift in high-dimensional data using Euclidean or cosine distances can misattribute changes due to arbitrary parametrizations.
method Introducing the Fubini Study metric to identify representations that differ only by gauge transformations.
result The Fubini Study metric isolates intrinsic evolution by remaining invariant under gauge-induced fluctuations, providing a diagnostic for meaningful structural changes.
Geometrically characterizes virtual nonlinear nonholonomic constraints using symplectic methods.
problem Characterizing virtual nonlinear nonholonomic constraints geometrically.
method Geometric characterization using symplectic structures and Chetaev equations.
result A unique control law exists to satisfy virtual constraints, and closed-loop dynamics are projections of uncontrolled dynamics.
The paper tracks patient recovery using graphs of joint movement data.
problem Tracking individual patient recovery trajectories in physical therapy.
method Bayesian learning of Random Geometric Graphs from joint movement data.
result Optimal exercise routines can be recommended based on patient recovery data.
The paper defines symmetries in no-arbitrage markets.
problem Characterizing transformations preserving no-arbitrage.
method Geometric formalization in discrete time models.
result Local characterization of no-arbitrage symmetries.
Signature tensors uniquely identify ODE solutions.
problem Identifying ODE solutions from signature tensors.
method Geometric theory of nonlinear systems of ODEs.
result Necessary and sufficient algebraic conditions for signature tensors to represent ODE solutions.
LLMs learn probability density functions in-context, showing distinct learning trajectories.
problem Density estimation of time series data in LLMs.
method Intensive Principal Component Analysis (InPCA) to visualize and analyze LLMs' learning dynamics.
result LLMs follow similar learning trajectories in a low-dimensional InPCA space, distinct from traditional methods.
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.
problem Robust representation learning of temporal dynamic interactions in robotics.
method Geometric approach using Procrustes distance and optimal transport for comparing interaction distributions.
result Metrics for assessing stability and comparing interaction learning algorithms.
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 …