乘法神经网络架构：近似中的局部性与正则性
A Multiplicative Neural Network Architecture: Locality and Regularity of Approximation
AI总结：
本文提出一种乘法神经网络架构，通过乘法交互实现近似，展示了其在局部性和正则性方面的改进表现。
摘要：
我们介绍了一种乘法神经网络架构，其中乘法交互构成了基本的表示形式，而不是作为加法模型中的辅助组件出现。我们为这种架构建立了通用近似定理，并分析了其在Bessel势空间中的近似性质，即局部性和正则性。为了补充理论结果，我们在具有尖锐过渡层或点wise失去高阶正则性的代表性目标上进行了数值实验。实验重点在于近似误差的空间结构以及对正则性敏感的量，特别是Zygmund型半范数的收敛性。结果表明，所提出的乘法架构产生的残差误差结构更紧密地与正则性降低的区域对齐，并在正则性敏感的度量中表现出更稳定的收敛性。这些结果表明，采用乘法表示形式对神经网络近似中的局部性和正则性行为有实际影响，为架构设计与近似函数分析属性之间提供了直接联系。
英文摘要：
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 has concrete implications for the localization and regularity behavior of neural network approximations, providing a direct connection between architectural design and analytical properties of the approximating functions.
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