On the gradient conjecture at infinity: Finite spherical length for gradient trajectories of definable potentials
Abstract
We prove that for every open $ U \subseteq \mathbb R^n $ the spherical projection of every gradient trajectory θ(t), t ≥ 0, of a definable C1-potential F : U → ℝ has finite length and, in particular, converges (directional convergence). In other words, every gradient trajectory of a definable C1-potential has finite total angular motion. Here, definable means definable in an arbitrary o-minimal structure expanding the real field. Notably, the gradient trajectory in this result is an arbitrary solution of the gradient flow differential equation (whose solutions may not be unique) with no further assumptions: The gradient trajectory may be unbounded and diverge to infinity or it may be convergent. The established theorem therefore provides affirmative answers to both the gradient conjecture at infinity for definable potentials and the definable Thom gradient conjecture. We apply our theorem to the training of deep neural networks (DNNs) to conclude that, in the case of definable and continuously differentiable activation and loss functions, the spherical projection of the DNN training flow must always converge (directional convergence).
Related
- Dynamical Groups and Spherical Potentials in Classical Mechanics
- The spherical cap conjecture for immersed disks
- Spherical-Design Oracles for Zeroth-Order Optimisation: Deterministic, Variance-Free Gradient Estimation
- Pólya's conjecture for Neumann eigenvalues of spherical shells
- Muon meets Tamed Langevin: Momentum Preconditioning beyond Convex and gradient-Lipschitz Potentials
BibTeX
@misc{do2026gradient,
title = {{On the gradient conjecture at infinity: Finite spherical length for gradient trajectories of definable potentials}},
author = {Thang Do and Arnulf Jentzen and Benno Kuckuck},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.29211},
eprint = {2610.29211},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.29211},
}
RIS
TY - UNPB
TI - On the gradient conjecture at infinity: Finite spherical length for gradient trajectories of definable potentials
AU - Thang Do
AU - Arnulf Jentzen
AU - Benno Kuckuck
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2610.29211
UR - https://arxiv.org/abs/2610.29211
AB - We prove that for every open $ U \subseteq \mathbb R^n $ the spherical projection of every gradient trajectory θ(t), t ≥ 0, of a definable C^1-potential F : U → ℝ has finite length and, in particular, converges (directional convergence). In other words, every gradient trajectory of a definable C^1-potential has finite total angular motion. Here, definable means definable in an arbitrary o-minimal structure expanding the real field. Notably, the gradient trajectory in this result is an arbitrary solution of the gradient flow differential equation (whose solutions may not be unique) with no further assumptions: The gradient trajectory may be unbounded and diverge to infinity or it may be convergent. The established theorem therefore provides affirmative answers to both the gradient conjecture at infinity for definable potentials and the definable Thom gradient conjecture. We apply our theorem to the training of deep neural networks (DNNs) to conclude that, in the case of definable and continuously differentiable activation and loss functions, the spherical projection of the DNN training flow must always converge (directional convergence).
ER -