{"ID":2828246,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2512.15992","arxiv_id":"2512.15992","title":"Time-Frequency Analysis for Neural Networks","abstract":"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.","short_abstract":"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 w...","url_abs":"https://arxiv.org/abs/2512.15992","url_pdf":"https://arxiv.org/pdf/2512.15992v2","authors":"[\"Ahmed Abdeljawad\",\"Elena Cordero\"]","published":"2025-12-17T21:51:51Z","proceeding":"math.NA","tasks":"[\"math.NA\",\"cs.IT\",\"cs.LG\"]","methods":"[]","has_code":false}