# Cluster-based classification with neural ODEs via control

arXiv: https://arxiv.org/abs/2312.13807 · HTML: https://mlsanity.com/paper/2312.13807/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 27 of its 54 references that are in the map.
- First posted on arXiv 21 Dec 2023.

## Builds on (most cited first, 27 of 27)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Understanding deep learning requires rethinking generalization](https://mlsanity.com/paper/1611.03530/) (2016)
- [Benign Overfitting in Linear Regression](https://mlsanity.com/paper/1906.11300/) (2019)
- [Stable Architectures for Deep Neural Networks](https://mlsanity.com/paper/1705.03341/) (2017)
- [The Power of Depth for Feedforward Neural Networks](https://mlsanity.com/paper/1512.03965/) (2015)
- [Dissecting Neural ODEs](https://mlsanity.com/paper/2002.08071/) (2020)
- [The emergence of clusters in self-attention dynamics](https://mlsanity.com/paper/2305.05465/) (2023)
- [Identity Matters in Deep Learning](https://mlsanity.com/paper/1611.04231/) (2016)
- [ResNet with one-neuron hidden layers is a Universal Approximator](https://mlsanity.com/paper/1806.10909/) (2018)
- [Maximum Principle Based Algorithms for Deep Learning](https://mlsanity.com/paper/1710.09513/) (2017)
- [Small ReLU networks are powerful memorizers: a tight analysis of memorization capacity](https://mlsanity.com/paper/1810.07770/) (2018)
- [Deep Learning via Dynamical Systems: An Approximation Perspective](https://mlsanity.com/paper/1912.10382/) (2019)
- [A Mean-Field Optimal Control Formulation of Deep Learning](https://mlsanity.com/paper/1807.01083/) (2018)
- [Neural ODE control for classification, approximation and transport](https://mlsanity.com/paper/2104.05278/) (2021)
- [Universal approximation property of invertible neural networks](https://mlsanity.com/paper/2204.07415/) (2022)
- [Deep neural networks, generic universal interpolation, and controlled ODEs](https://mlsanity.com/paper/1908.07838/) (2019)
- [Interpolation, Approximation and Controllability of Deep Neural Networks](https://mlsanity.com/paper/2309.06015/) (2023)
- [A Measure Theoretical Approach to the Mean-field Maximum Principle for Training NeurODEs](https://mlsanity.com/paper/2107.08707/) (2021)
- [Turnpike in optimal control of PDEs, ResNets, and beyond](https://mlsanity.com/paper/2202.04097/) (2022)
- [Control of neural transport for normalizing flows](https://mlsanity.com/paper/2307.07817/) (2023)
- [A Mathematical Framework for Learning Probability Distributions](https://mlsanity.com/paper/2212.11481/) (2022)
- [Sparsity in long-time control of neural ODEs](https://mlsanity.com/paper/2102.13566/) (2021)
- [Learning on Manifolds: Universal Approximations Properties using Geometric Controllability Conditions for Neural ODEs](https://mlsanity.com/paper/2305.08849/) (2023)
- [Interpolation and approximation via Momentum ResNets and Neural ODEs](https://mlsanity.com/paper/2110.08761/) (2021)
- [Variational formulations of ODE-Net as a mean-field optimal control problem and existence results](https://mlsanity.com/paper/2303.05924/) (2023)
- [Memorization Capacity of Neural Networks with Conditional Computation](https://mlsanity.com/paper/2303.11247/) (2023)

## Built on by (most cited first, 4 of 4)

- [Interplay between depth and width for interpolation in neural ODEs](https://mlsanity.com/paper/2401.09902/) (2024)
- [Exact Sequence Interpolation with Transformers](https://mlsanity.com/paper/2502.02270/) (2025)
- [Machine Learning and Control: Foundations, Advances, and Perspectives](https://mlsanity.com/paper/2510.03303/) (2025)
- [Constructive interpolation and generalization rates for neural ODEs: a control perspective](https://mlsanity.com/paper/2606.00469/) (2026)

## Abstract (arXiv)

We address binary classification using neural ordinary differential equations from the perspective of simultaneous control of N data points. We consider a single-neuron architecture with parameters fixed as piecewise constant functions of time. In this setting, the model complexity can be quantified by the number of control switches. Previous work has shown that classification can be achieved using a point-by-point strategy that requires O(N) switches. We propose a new control method that classifies any arbitrary dataset by sequentially steering clusters of d points, thereby reducing the complexity to O(N/d) switches. The optimality of this result, particularly in high dimensions, is supported by some numerical experiments. Our complexity bound is sufficient but often conservative because same-class points tend to appear in larger clusters, simplifying classification. This motivates studying the probability distribution of the number of switches required. We introduce a simple control method that imposes a collinearity constraint on the parameters, and analyze a worst-case scenario where both classes have the same size and all points are i.i.d. Our results highlight the benefits of high-dimensional spaces, showing that classification using constant controls becomes more probable as d increases.