arXiv:2605.21451v1 Announce Type: new Abstract: 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 s
The proliferation of complex neural network architectures across diverse applications necessitates a deeper theoretical understanding of their capabilities and limitations beyond empirical observation.
A robust approximation theory provides the mathematical foundations for designing more efficient and reliable AI systems, optimizing architectural choices, and ensuring predictable performance.
The ongoing evolution from qualitative universal approximation theorems to quantitative theories of approximation rates and parameter efficiency will significantly refine neural network development paradigms.
- · AI researchers
- · Deep learning practitioners
- · Hardware manufacturers (for optimized architectures)
- · Academia
- · Empirical-only AI development approaches
Improved understanding of neural network limitations and capabilities.
Development of novel, more efficient neural network architectures based on theoretical insights.
Accelerated AI progress due to principled design rather than trial-and-error, potentially affecting various sectors.
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Read at arXiv cs.LG