Time-Frequency Analysis for Neural Networks
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866910120156332032 |
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| author | Abdeljawad, Ahmed Cordero, Elena |
| author_facet | Abdeljawad, Ahmed Cordero, Elena |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_15992 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Time-Frequency Analysis for Neural Networks Abdeljawad, Ahmed Cordero, Elena Numerical Analysis Information Theory Machine Learning 41A25, 41A46, 41A65, 46E35, 68T07, 68T05 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. |
| title | Time-Frequency Analysis for Neural Networks |
| topic | Numerical Analysis Information Theory Machine Learning 41A25, 41A46, 41A65, 46E35, 68T07, 68T05 |
| url | https://arxiv.org/abs/2512.15992 |