Sharp Lower Bounds for Linearized ReLU^k Approximation on the Sphere
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.
Code (0)
등록된 구현이 없습니다.
Similar Papers 제목 키워드 기반
Sharp Lower Bounds on the Approximation Rate of Shallow Neural Networks
We consider the approximation rates of shallow neural networks with respect to the variation norm. Upper bounds on these rates have been established for sigmoidal and ReLU activation functions, but it has remained an imp…
Sharp Lower Bounds on Interpolation by Deep ReLU Neural Networks at Irregularly Spaced Data
We study the interpolation power of deep ReLU neural networks. Specifically, we consider the question of how efficiently, in terms of the number of parameters, deep ReLU networks can interpolate values at $N$ datapoints …
MemorizationSharp Bounds on the Approximation Rates, Metric Entropy, and $n$-widths of Shallow Neural Networks
In this article, we study approximation properties of the variation spaces corresponding to shallow neural networks with a variety of activation functions. We introduce two main tools for estimating the metric entropy, a…
Error bounds for approximations with deep ReLU neural networks in $W^{s,p}$ norms
We analyze approximation rates of deep ReLU neural networks for Sobolev-regular functions with respect to weaker Sobolev norms. First, we construct, based on a calculus of ReLU networks, artificial neural networks with R…
Covering Numbers for Deep ReLU Networks with Applications to Function Approximation and Nonparametric Regression
Covering numbers of families of (deep) ReLU networks have been used to characterize their approximation-theoretic performance, upper-bound the prediction error they incur in nonparametric regression, and quantify their c…
Quantizationregression