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[2602.12482] Geometric separation and constructive universal approximation with two hidden layers
Summary
This paper presents a geometric construction of neural networks capable of separating disjoint compact subsets in R^n, demonstrating a universal approximation theorem for networks with two hidden layers using sigmoidal or ReLU activations.
Why It Matters
The findings contribute to the understanding of neural network architecture and approximation capabilities, particularly in machine learning. By establishing a constructive universal approximation theorem, this research can influence the design of more efficient neural networks for various applications.
Key Takeaways
- Introduces a geometric method for neural network construction.
- Proves that two hidden layer networks can approximate any continuous function on compact sets.
- Utilizes both sigmoidal and ReLU activation functions in the approximation process.
- Simplifies the approximation for finite compact sets to a depth-2 result.
- Enhances theoretical foundations for neural network capabilities in machine learning.
Computer Science > Machine Learning arXiv:2602.12482 (cs) [Submitted on 12 Feb 2026] Title:Geometric separation and constructive universal approximation with two hidden layers Authors:Chanyoung Sung View a PDF of the paper titled Geometric separation and constructive universal approximation with two hidden layers, by Chanyoung Sung View PDF HTML (experimental) Abstract:We give a geometric construction of neural networks that separate disjoint compact subsets of $\Bbb R^n$, and use it to obtain a constructive universal approximation theorem. Specifically, we show that networks with two hidden layers and either a sigmoidal activation (i.e., strictly monotone bounded continuous) or the ReLU activation can approximate any real-valued continuous function on an arbitrary compact set $K\subset\Bbb R^n$ to any prescribed accuracy in the uniform norm. For finite $K$, the construction simplifies and yields a sharp depth-2 (single hidden layer) approximation result. Subjects: Machine Learning (cs.LG); Classical Analysis and ODEs (math.CA) MSC classes: 41A46, 68T07, 54D15 Cite as: arXiv:2602.12482 [cs.LG] (or arXiv:2602.12482v1 [cs.LG] for this version) https://doi.org/10.48550/arXiv.2602.12482 Focus to learn more arXiv-issued DOI via DataCite (pending registration) Submission history From: Chanyoung Sung [view email] [v1] Thu, 12 Feb 2026 23:46:11 UTC (114 KB) Full-text links: Access Paper: View a PDF of the paper titled Geometric separation and constructive universal approximati...