用时频分析证明神经网络可高效逼近函数，效果优于传统ReLU网络。
Time-Frequency Analysis for Neural Networks
- 基于时频窗口的神经元结构，结合加权调制空间理论分析
- 在有界域上达到N^{-1/2}的独立维数逼近率，常数可显式控制
- 适用于高维函数逼近，对频域敏感任务特别有效
我们利用时频分析工具，为浅层神经网络建立定量逼近理论。在加权调制空间 $M^{p,q}_m(oldsymbol{R}^d)$ 中，证明了当网络单元结合标准激活与局域时频窗时，可在 Sobolev 范数 $W^{n,r}(Ω)$ 下实现维度无关的逼近率。主要结果表明：对 $f o M^{p,q}_m(oldsymbol{R}^d)$，在有界域上可达到 \\[ \|f - f_N\|_{W^{n,r}(Ω)} \lesssim N^{-1/2}\,\|f\|_{M^{p,q}_m(oldsymbol{R}^d)} \\[，且所有常数可显式控制。进一步在 $oldsymbol{R}^d$ 上获得全局逼近定理，涵盖 Feichtinger 代数、Fourier-Lebesgue 空间和 Barron 空间。一维与二维数值实验表明，基于调制的网络在 Sobolev 逼近上显著优于标准 ReLU 网络，与理论估计一致。
原文摘要 · Abstract (English)
We develop a quantitative approximation theory for shallow neural networks using tools from time-frequency analysis. Working in weighted modulation spaces $M^{p,q}_m(\mathbf{R}^{d})$, we prove dimension-independent approximation rates in Sobolev norms $W^{n,r}(Ω)$ for networks whose units combine standard activations with localized time-frequency windows. Our main result shows that for $f \in M^{p,q}_m(\mathbf{R}^{d})$ one can achieve \[ \|f - f_N\|_{W^{n,r}(Ω)} \lesssim N^{-1/2}\,\|f\|_{M^{p,q}_m(\mathbf{R}^{d})}, \] on bounded domains, with explicit control of all constants. We further obtain global approximation theorems on $\mathbf{R}^{d}$ using weighted modulation dictionaries, and derive consequences for Feichtinger's algebra, Fourier-Lebesgue spaces, and Barron spaces. Numerical experiments in one and two dimensions confirm that modulation-based networks achieve substantially better Sobolev approximation than standard ReLU networks, consistent with the theoretical estimates.
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