# Configuration-Dependent Lower Bounds for Approximation by Shallow ReLUk Networks on the Sphere

arXiv: https://arxiv.org/abs/2510.04060 · HTML: https://mlsanity.com/paper/2510.04060/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 39 references that are in the map.
- First posted on arXiv 5 Oct 2025.

## Builds on (most cited first, 9 of 9)

- [Error bounds for approximations with deep ReLU networks](https://mlsanity.com/paper/1610.01145/) (2016)
- [Breaking the Curse of Dimensionality with Convex Neural Networks](https://mlsanity.com/paper/1412.8690/) (2014)
- [Sharp Bounds on the Approximation Rates, Metric Entropy, and n-widths of Shallow Neural Networks](https://mlsanity.com/paper/2101.12365/) (2021)
- [Approximation by Combinations of ReLU and Squared ReLU Ridge Functions with  ℓ1  and  ℓ0  Controls](https://mlsanity.com/paper/1607.07819/) (2016)
- [Rates of Approximation by ReLU Shallow Neural Networks](https://mlsanity.com/paper/2307.12461/) (2023)
- [Optimal Approximation of Zonoids and Uniform Approximation by Shallow Neural Networks](https://mlsanity.com/paper/2307.15285/) (2023)
- [Expressivity and Approximation Properties of Deep Neural Networks with ReLUk Activation](https://mlsanity.com/paper/2312.16483/) (2023)
- [Approximation Rates for Shallow ReLUk Neural Networks on Sobolev Spaces via the Radon Transform](https://mlsanity.com/paper/2408.10996/) (2024)
- [Tractability of approximation by general shallow networks](https://mlsanity.com/paper/2308.03230/) (2023)

## Built on by (most cited first, 1 of 1)

- [Shallow neural network approximation in mixed Sobolev spaces](https://mlsanity.com/paper/2609.05263/) (2026)

## Abstract (arXiv)

We establish two related but logically distinct results for shallow ReLUk neural networks on the unit sphere SSd. First, for an arbitrary set of inner neural-network parameters, the best L2(SSd) approximation of a fixed target function with smoothness r>tfracd+2k+12 admits an asymptotic lower bound given by a constant multiple of n-1/2underlinehk+1/2, where underlineh denotes the antipodal separation distance of the normalized inner-parameter set. This lower bound depends explicitly on the parameter configuration through underlineh and applies without additional assumptions on the parameters. Second, for antipodally quasi-uniform parameters, underlinehsimeq n-1/d, and the lower bound establishes the exact saturation order n^-fracd+2k+12d for such parameter families: a target function with regularity greater than fracd+2k+12 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 ReLUk network spaces can outperform finite elements of the same degree, this advantage is intrinsically limited.