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[2602.06374] A Multiplicative Neural Network Architecture: Locality and Regularity of Approximation
Summary
This paper presents a novel multiplicative neural network architecture that emphasizes multiplicative interactions for function approximation, establishing a universal approximation theorem and analyzing its properties through numerical experiments.
Why It Matters
The research introduces a new architectural design for neural networks that could enhance their performance in approximating complex functions. By focusing on multiplicative interactions, it provides insights into how architecture influences approximation behavior, which is critical for advancing machine learning applications.
Key Takeaways
- Introduces a multiplicative neural network architecture.
- Establishes a universal approximation theorem for the new architecture.
- Analyzes approximation properties in terms of locality and regularity.
- Demonstrates improved error structures in regions of reduced regularity.
- Connects architectural design to analytical properties of approximating functions.
Mathematics > Functional Analysis arXiv:2602.06374 (math) [Submitted on 6 Feb 2026 (v1), last revised 15 Feb 2026 (this version, v2)] Title:A Multiplicative Neural Network Architecture: Locality and Regularity of Approximation Authors:Hee-Sun Choi, Beom-Seok Han View a PDF of the paper titled A Multiplicative Neural Network Architecture: Locality and Regularity of Approximation, by Hee-Sun Choi and Beom-Seok Han View PDF HTML (experimental) Abstract:We introduce a multiplicative neural network architecture in which multiplicative interactions constitute the fundamental representation, rather than appearing as auxiliary components within an additive model. We establish a universal approximation theorem for this architecture and analyze its approximation properties in terms of locality and regularity in Bessel potential spaces. To complement the theoretical results, we conduct numerical experiments on representative targets exhibiting sharp transition layers or pointwise loss of higher-order regularity. The experiments focus on the spatial structure of approximation errors and on regularity-sensitive quantities, in particular, the convergence of Zygmund-type seminorms. The results show that the proposed multiplicative architecture yields residual error structures that are more tightly aligned with regions of reduced regularity and exhibit more stable convergence in regularity-sensitive metrics. These results demonstrate that adopting a multiplicative representation format ha...
