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About this paper
Chanyoung Sung
Chanyoung Sung presents a constructive universal approximation theorem for neural networks featuring two hidden layers, offering explicit methods to approximate continuous functions on compact sets using both sigmoidal and ReLU activations. The work establishes how these networks can geometrically separate disjoint compact sets, addressing the non-constructive aspect of classical approximation proofs.
Key insights
- Increasing network depth to two hidden layers (depth 3) provides the necessary 'geometric flexibility' to constructively implement fundamental topological separation and universal approximation.
- The second hidden layer explicitly functions as an 'aggregator' or 'selector' of the first layer's outputs, enabling the construction of more complex functions like those required for Urysohn-style set separation.
- A unified proof template successfully demonstrates constructive universal approximation for two distinct classes of activation functions: strictly monotone bounded continuous (sigmoidal) and Rectified Linear Units (ReLU).
Summary by alphaXiv.
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