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Volume 4, Issue 2
Cluster-Based Classification with Neural ODEs via Control

Antonio Álvarez-Lόpez, Rafael Orive-Illera & Enrique Zuazua

J. Mach. Learn. , 4 (2025), pp. 128-156.

Published online: 2025-06

[An open-access article; the PDF is free to any online user.]

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  • 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.

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@Article{JML-4-128, author = {Álvarez-Lόpez , AntonioOrive-Illera , Rafael and Zuazua , Enrique}, title = {Cluster-Based Classification with Neural ODEs via Control}, journal = {Journal of Machine Learning}, year = {2025}, volume = {4}, number = {2}, pages = {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.

}, issn = {2790-2048}, doi = {https://doi.org/10.4208/jml.241114}, url = {http://global-sci.org/intro/article_detail/jml/24145.html} }
TY - JOUR T1 - Cluster-Based Classification with Neural ODEs via Control AU - Álvarez-Lόpez , Antonio AU - Orive-Illera , Rafael AU - Zuazua , Enrique JO - Journal of Machine Learning VL - 2 SP - 128 EP - 156 PY - 2025 DA - 2025/06 SN - 4 DO - http://doi.org/10.4208/jml.241114 UR - https://global-sci.org/intro/article_detail/jml/24145.html KW - Classification, Complexity, Neural ODEs, Separability, Controllability. AB -

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.

Álvarez-Lόpez , AntonioOrive-Illera , Rafael and Zuazua , Enrique. (2025). Cluster-Based Classification with Neural ODEs via Control. Journal of Machine Learning. 4 (2). 128-156. doi:10.4208/jml.241114
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