Geometric separation and constructive universal approximation with two hidden layers

• URL: http://arxiv.org/abs/2602.12482v1
• Date: Thu, 12 Feb 2026 23:46:11 GMT
• Title: Geometric separation and constructive universal approximation with two hidden layers
• Authors: Chanyoung Sung,
• Abstract summary: 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.<n>For finite $K$, the construction simplifies and yields a sharp depth-2 (single hidden layer) approximation result.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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