Configuration-Dependent Lower Bounds for Approximation by Shallow ReLU$^k$ Networks on the Sphere

arXiv:2510.04060v3 Announce Type: replace-cross Abstract: We establish two related but logically distinct results for shallow ReLU$^k$ neural networks on the unit sphere $\SS^d$. First, for an arbitrary set of inner neural-network parameters, the best $\mathcal{L}^2(\SS^d)$ approximation of a fixed target function with smoothness $r>\tfrac{d+2k+1}{2}$ admits an asymptotic lower bound given by a constant multiple of $n^{-1/2}\underline{h}^{k+1/2}$, where $\underline{h}$ denotes the antipodal separation distance of the normalized inner-parameter set. This lower bound depends explicitly on the parameter configuration through $\underline{h}$ and applies without additional assumptions on the parameters. Second, for antipodally quasi-uniform parameters, $\underline{h}\simeq n^{-1/d}$, and the lower bound establishes the exact saturation order $n^{-\frac{d+2k+1}{2d}}$ for such parameter families: a target function with regularity greater than $\frac{d+2k+1}{2}$ and satisfying the required parity condition can be approximated at this rate, whereas approximation at any strictly faster rate forces the target function to be zero. Our results therefore place linearized neural-network approximation within the classical saturation framework and show that, although ReLU$^k$ network spaces can outperform finite elements of the same degree, this advantage is intrinsically limited.

Abstract:We establish two related but logically distinct results for shallow ReLU$^k$ neural networks on the unit sphere $\SS^d$. First, for an arbitrary set of inner neural-network parameters, the best $\mathcal{L}^2(\SS^d)$ approximation of a fixed target function with smoothness $r>\tfrac{d+2k+1}{2}$ admits an asymptotic lower bound given by a constant multiple of $n^{-1/2}\underline{h}^{k+1/2}$, where $\underline{h}$ denotes the antipodal separation distance of the normalized inner-parameter set. This lower bound depends explicitly on the parameter configuration through $\underline{h}$ and applies without additional assumptions on the parameters. Second, for antipodally quasi-uniform parameters, $\underline{h}\simeq n^{-1/d}$, and the lower bound establishes the exact saturation order $n^{-\frac{d+2k+1}{2d}}$ for such parameter families: a target function with regularity greater than $\frac{d+2k+1}{2}$ and satisfying the required parity condition can be approximated at this rate, whereas approximation at any strictly faster rate forces the target function to be zero. Our results therefore place linearized neural-network approximation within the classical saturation framework and show that, although ReLU$^k$ network spaces can outperform finite elements of the same degree, this advantage is intrinsically limited.

Numerical Analysis (math.NA); Machine Learning (cs.LG)

arXiv:2510.04060 [math.NA]

(or arXiv:2510.04060v3 [math.NA] for this version)

https://doi.org/10.48550/arXiv.2510.04060 Focus to learn more arXiv-issued DOI via DataCite

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Configuration-Dependent Lower Bounds for Approximation by Shallow ReLU$^k$ Networks on the Sphere

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