{"ID":2860219,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2510.04060","arxiv_id":"2510.04060","title":"Sharp Lower Bounds for Linearized ReLU^k Approximation on the Sphere","abstract":"We prove a saturation theorem for linearized shallow ReLU$^k$ neural networks on the unit sphere $\\mathbb S^d$. For any antipodally quasi-uniform set of centers, if the target function has smoothness $r\u003e\\tfrac{d+2k+1}{2}$, then the best $\\mathcal{L}^2(\\mathbb S^d)$ approximation cannot converge faster than order $n^{-\\frac{d+2k+1}{2d}}$. This lower bound matches existing upper bounds, thereby establishing the exact saturation order $\\tfrac{d+2k+1}{2d}$ for such networks. Our results place linearized neural-network approximation firmly within the classical saturation framework and show that, although ReLU$^k$ networks outperform finite elements under equal degrees $k$, this advantage is intrinsically limited.","short_abstract":"We prove a saturation theorem for linearized shallow ReLU$^k$ neural networks on the unit sphere $\\mathbb S^d$. For any antipodally quasi-uniform set of centers, if the target function has smoothness $r\u003e\\tfrac{d+2k+1}{2}$, then the best $\\mathcal{L}^2(\\mathbb S^d)$ approximation cannot converge faster than order $n^{-\\...","url_abs":"https://arxiv.org/abs/2510.04060","url_pdf":"https://arxiv.org/pdf/2510.04060v2","authors":"[\"Tong Mao\",\"Jinchao Xu\"]","published":"2025-10-05T06:47:24Z","proceeding":"math.NA","tasks":"[\"math.NA\",\"cs.LG\"]","methods":"[]","has_code":false}