揭示了球面上ReLU^k网络逼近的理论极限，证明其收敛速度有严格下界。
Sharp Lower Bounds for Linearized ReLU^k Approximation on the Sphere
- 针对球面单位上线性化浅层ReLU^k网络，建立逼近误差下界
- 当函数光滑度r > (d+2k+1)/2时，最优逼近误差不快于n^(-(d+2k+1)/(2d))
- 首次明确该类网络的饱和阶为(d+2k+1)/(2d)，适用于理论研究者
本文证明了在单位球面$/mathbb S^d$上，对任意对称准均匀中心集，若目标函数光滑度$r > \frac{d+2k+1}{2}$，则线性化浅层ReLU$^k$神经网络在$\/mathcal{L}^2(\/mathbb S^d)$范数下的最佳逼近误差收敛速度不可能超过阶数$n^{-\frac{d+2k+1}{2d}}$。该下界与已有上界一致，从而确立了此类网络的精确饱和阶为$\frac{d+2k+1}{2d}$。研究将神经网络逼近置于经典饱和理论框架中，表明尽管ReLU$^k$网络在相同次数$k$下优于有限元方法，但其优势是内在受限的。
原文摘要 · Abstract (English)
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 $r>\tfrac{d+2k+1}{2}$, 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.
Thank you to arXiv for use of its open access interoperability. PaperDance 不是 arXiv 官方产品；中文卡片由大模型生成，请以原文为准。