# Control in the spaces of ensembles of points

arXiv: https://arxiv.org/abs/1907.00905 · HTML: https://mlsanity.com/paper/1907.00905/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 1 Jul 2019.

## Built on by (most cited first, 9 of 9)

- [Measure-to-measure interpolation using Transformers](https://mlsanity.com/paper/2411.04551/) (2024)
- [Deep Learning Approximation of Diffeomorphisms via Linear-Control Systems](https://mlsanity.com/paper/2110.12393/) (2021)
- [Normalizing flows as approximations of optimal transport maps via linear-control neural ODEs](https://mlsanity.com/paper/2311.01404/) (2023)
- [Control Theoretic Approach to Fine-Tuning and Transfer Learning](https://mlsanity.com/paper/2404.11013/) (2024)
- [Geometric Foundations of Tuning without Forgetting in Neural ODEs](https://mlsanity.com/paper/2509.03474/) (2025)
- [Control Disturbance Rejection in Neural ODEs](https://mlsanity.com/paper/2509.18034/) (2025)
- [Constructive interpolation and generalization rates for neural ODEs: a control perspective](https://mlsanity.com/paper/2606.00469/) (2026)
- [Privacy Preserving Gossip Learning](https://mlsanity.com/paper/2609.14778/) (2026)
- [Deep learning from a control vantage point](https://mlsanity.com/paper/2610.14385/) (2026)

## Abstract (arXiv)

We study the controlled dynamics of the it ensembles of points of a Riemannian manifold M. Parameterized ensemble of points of M is the image of a continuous map γ:Theta → M, where Theta is a compact set of parameters. The dynamics of ensembles is defined by the action γ(θ) mapsto Pt(γ(θ)) of the semigroup of diffeomorphisms Pt:M → M,  t ∈ ℝ, generated by the controlled equation dotx=f(x,u(t)) on M. Therefore any control system on M defines a control system on (generally infinite-dimensional) space E_Theta(M) of the ensembles of points. We wish to establish criteria of controllability for such control systems. As in our previous work ([1]) we seek to adapt the Lie-algebraic approach of geometric control theory to the infinite-dimensional setting. We study the case of finite ensembles and prove genericity of exact controllability property for them. We also find sufficient approximate controllability criterion for continual ensembles and prove a result on motion planning in the space of flows on M. We discuss the relation of the obtained controllability criteria to various versions of Rashevsky-Chow theorem for finite- and infinite-dimensional manifolds.