Neural Network

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Approximation Theory for Neural Networks: Old and New
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May 20, 2026 · cs.LGJ/K move · Enter open · S save

Approximation Theory for Neural Networks: Old and New

Soumendu Sundar Mukherjee, Himasish Talukdar

Statistics and Mathematics Unit, Indian Statistical Institute, Kolkata

Universal approximation theorems provide a mathematical explanation for the expressive power of neural networks. They assert that, under mild conditions on the activation function, feedforward neural networks are dense in broad function classes, such as continuous functions on compact subsets of

\mathbb{R}^d

L^p

spaces, or Sobolev spaces. Over the past four decades, these qualitative universality results have evolved into a rich quantitative theory addressing approximation rates, parameter efficiency, and the role of architectural features such as depth and width. This survey presents several glimpses into this theory. We review classical density results for single-hidden-layer networks, as well as quantitative bounds that relate approximation error to network size and smoothness assumptions on target functions. Particular emphasis is placed on depth--width trade-offs and on results demonstrating that deeper architectures can achieve superior parameter efficiency for structured function classes. In addition to standard feedforward neural networks, we also review recent developments on Kolmogorov--Arnold Networks (KANs), which offer an alternative architectural paradigm and whose approximation-theoretic properties have begun to attract significant theoretical attention.

Neural-Network Approximation Approximation Neural Network Neural Architectures Kolmogorov-Arnold Networks Feed-Forward Network Age

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