Function Approximation

Latest papers

32 papers3 papers in the last four weeks, down 25% on the four weeks before. 0.0% of all new papers.

Configuration-Dependent Lower Bounds for Approximation by Shallow ReLU $^k$ Networks on the Sphere
Oct 5, 2025Tong Mao, Jinchao XuShallow Neural Networks Neural Network Approximation Theory
Shenzhen Loop Area Institute, Shenzhen 518000, P. R. China · King Abdullah University of Science and Technology, Thuwal 23955, Saudi Arabia.
Uniform Approximation of Functions with Asymmetric Growth and Decay by Deep Weighted Polynomials
Jun 26, 2025Kingsley Yeon, Steven B. DamelinNeural Network Approximation Theory Function Approximation
Department of Statistics and CCAM, University of Chicago, Chicago, 60637, IL, USA · Department of Mathematics, ZBMATH-OPEN, Hermann-von-Helmholtz-Platz 1, Eggenstein-Leopoldshafen, 76344, Germany

Oct 5, 2025J/K move · Enter open · S save

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

Tong Mao, Jinchao Xu

Shenzhen Loop Area Institute, Shenzhen 518000, P. R. China · King Abdullah University of Science and Technology, Thuwal 23955, Saudi Arabia.

Shallow Neural Networks Neural Network Approximation Theory Neural Network Expressivity Function Approximation

Open paper AskPDF arXiv

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.

Latest papers

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

Uniform Approximation of Functions with Asymmetric Growth and Decay by Deep Weighted Polynomials

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