发表机构
IRMA（IRMA）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出广义Radon变换的纯分布框架，解决无限宽神经网络标准激活函数的谱奇异性问题，证明Barron函数与其最优权重测度间存在精确线性等距映射。
AI 中文摘要
无限宽浅层神经网络的理论基础严重依赖连续积分表示和Barron空间。近来，调和分析——特别是Radon变换和Ridgelet变换——已成为逆转这些表示并计算最优网络权重的有力工具。然而，一个主要的分析瓶颈仍然存在：标准神经网络激活函数在频率原点处表现出严重的谱奇异性。为绕过这种发散，现有框架要么将理论限制于特定激活函数族，要么数学上约去网络的仿射分量，这固有地限制了它们的实际应用范围。本文中，我们通过引入针对广义Radon变换R_σ的纯分布框架克服这些局限，该框架作用于一类广泛的缓增分布激活函数。通过定义正则化谱公式g(ρ)=(iρ)^α σ(ρ)，我们严格吸收原点奇异性而不截断基础函数空间。基于此精确重构，我们证明在激活函数满足确定奇偶性假设时，Barron函数与其最优权重测度之间存在精确的线性等距映射。
英文摘要
The theoretical foundation of infinite-width shallow neural networks relies heavily on continuous integral representations and Barron spaces. Recently, harmonic analysis-specifically the Radon and Ridgelet transforms-has emerged as a powerful tool to invert these representations and compute the optimal network weights. However, a major analytical bottleneck remains: standard neural network activation functions exhibit severe spectral singularities at the frequency origin. To bypass this divergence, existing frameworks either restrict the theory to specific activation families or mathematically quotient out the network's affine components, which inherently limits their practical scope. In this paper, we overcome these limitations by introducing a purely distributional framework for the generalized Radon transform R $σ$ that operates on a broad class of tempered distribution activation functions. By defining a regularized spectrum formulation g($ρ$) = (i$ρ$) $α$ $σ$($ρ$), we rigorously absorb the origin singularities without truncating the underlying functional space. Building upon this exact reconstruction, we show that under a definite parity assumption on the activation, there exists an exact linear isometry between Barron functions and their optimal weight measures.
