On Explicit Super-Expressive Approximation for Neural Networks

2026-07-07 · Source: Machine Learning · Field: Technology & Digital — Artificial Intelligence & Machine Learning · Depth: Expert, quick

Summary

This work investigates fixed-architecture neural network approximation using explicit parameter bounds and elementary activations. It addresses a gap in prior research, which lacked quantitative and non-asymptotic characterizations of parameter magnitude relative to approximation error. The authors introduce the Chinese Remainder Theorem as a constructive encoding mechanism to resolve this. For Lipschitz continuous functions on [0,1]^D, they construct a width-max{D,4}, depth-5 network with explicit parameter-error trade-offs. For Hölder-smooth functions in C^{r,γ}_A([0,1]^D), a fixed network of width max{2D, D+5N+1} and depth r + 9 achieves a parameter magnitude ℘ bounded by log₂ ℘=ℴ(ε⁻²ᴰ/(ᵣ+ɣ)log(1/ε)). This represents a dual result compared to parameter-bounded, architecture-unbounded paradigms.

Key takeaway

For AI scientists and machine learning researchers designing or analyzing neural network architectures for function approximation, this work offers a novel theoretical framework. You should consider how the Chinese Remainder Theorem can be applied to achieve explicit parameter bounds and predictable error trade-offs in fixed-architecture networks. This approach provides a quantitative understanding of network expressivity, moving beyond asymptotic characterizations.

Key insights

The Chinese Remainder Theorem enables explicit parameter bounds for super-expressive fixed-architecture neural networks.

Principles

• Fixed-architecture networks can achieve super-expressive approximation.
• Parameter magnitude characterization is crucial for approximation error.
• The Chinese Remainder Theorem offers a constructive encoding mechanism.

Method

Constructing fixed-width, fixed-depth neural networks using the Chinese Remainder Theorem to explicitly bound parameters for function approximation.

In practice

• Design networks with explicit parameter-error trade-offs.
• Apply CRT for constructive encoding in network design.

Topics

• Neural Networks
• Function Approximation
• Chinese Remainder Theorem
• Parameter Bounds
• Super-Expressive Networks
• Lipschitz Functions

Best for: Research Scientist, AI Scientist

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Editorial summary, takeaway, and curation by AIssential. Original article published by Machine Learning.