Year: 2025
Author: Antonio Álvarez-Lόpez, Rafael Orive-Illera, Enrique Zuazua
Journal of Machine Learning, Vol. 4 (2025), Iss. 2 : pp. 128–156
Abstract
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 $\mathcal{O}(N)$ switches. We propose a new control method that classifies any arbitrary dataset by sequentially steering clusters of $d$ points, thereby reducing the complexity to $\mathcal{O}(N/d)$ 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.
Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/jml.241114
Journal of Machine Learning, Vol. 4 (2025), Iss. 2 : pp. 128–156
Published online: 2025-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 29
Keywords: Classification Complexity Neural ODEs Separability Controllability.