mlsanityData to 14 Oct 2026

Optimal Neural Network Approximation via Empirical Least Squares with Deterministic Samples

arXiv 2608.06687 ↗First posted 7 Aug 2026math.NA cs.LG cs.NAcited by 0

Not on the map. Why?

Abstract (arXiv)

We develop a rigorous theory of discrete residual least-squares approximation for elliptic spectral equations mathfrak Lβ u=f using linearized ReLUk neural networks on the sphere, where mathfrak Lβ is a positive elliptic spectral multiplier of order β. Given a parameter set Thetan=θj^*j=1nsubsetmathbb Sd, we approximate u in the linearized network space Lnk(Thetan) by the discrete residual on the collocation points ηi^*i=1m beginequation* un,m∈argminvn∈ Lnk(Thetan)frac1msumi=1mleft(f(ηi^*)-mathfrak Lβ vn(ηi^*)right)2. endequation* With k>fracd-12+β, for antipodally quasi-uniform network parameter sets and any quasi-uniform collocation points with mgtrsim n, we prove that beginequation* |u-un,m|_mathcal Hβ(mathbb Sd)eqsim|f-mathfrak Lβ un,m|mathcal L2(mathbb Sd)lesssim n^-fracrd begincases |f|_mathcal Wr,p(mathbb Sd),&fracdp<r≤ fracd2,~p>2, |f|mathcal Hr(mathbb Sd),&r>fracd2. endcases endequation* We also establish a high-probability residual estimate, up to a logarithmic factor and an arbitrarily small smoothness loss, for i.i.d. uniformly distributed collocation points. The key analytical ingredient is a Bernstein inequality for linearized ReLUk network spaces. If underline h denotes the antipodal separation distance of the network parameters, then beginequation* |vn|mathcal Hr(mathbb Sd)lesssimunderline h-(r-s)|vn|mathcal Hs(mathbb Sd),qquad 0≤ s<r<k+tfrac12. endequation*

Key facts. No citing papers in the map yet (data through 14 Oct 2026). Builds on 19 of its 96 references that are in the map. First posted on arXiv 7 Aug 2026.

Builds on

19 of 96 references.

References · 19

  1. 2018Spherical CNNscited by 456
  2. 2014Breaking the Curse of Dimensionality with Convex Neural Networkscited by 331
  3. 2016Generalization Properties of Learning with Random Featurescited by 164
  4. 2021Sharp Bounds on the Approximation Rates, Metric Entropy, and n-widths of Shallow Neural Networkscited by 86
  5. 2016Approximation by Combinations of ReLU and Squared ReLU Ridge Functions with ℓ1 and ℓ0 Controlscited by 84
  6. 2019Spherical CNNs on Unstructured Gridscited by 79
  7. 2018A Priori Estimates of the Population Risk for Two-layer Neural Networkscited by 76
  8. 2020Representation formulas and pointwise properties for Barron functionscited by 62
  9. 2021Characterization of the Variation Spaces Corresponding to Shallow Neural Networkscited by 48
  10. 2020On the distance between two neural networks and the stability of learningcited by 46
  11. 2021Near-Minimax Optimal Estimation With Shallow ReLU Neural Networkscited by 41
  12. 2020Theory of Deep Convolutional Neural Networks II: Spherical Analysiscited by 22
7 more
  1. 2023An Extreme Learning Machine-Based Method for Computational PDEs in Higher Dimensionscited by 21
  2. 2019DeepSphere: towards an equivariant graph-based spherical CNNcited by 13
  3. 2020Kernel based analysis of massive datacited by 11
  4. 2017Function approximation with zonal function networks with activation functions analogous to the rectified linear unit functionscited by 8
  5. 2024Approximation Rates for Shallow ReLUk Neural Networks on Sobolev Spaces via the Radon Transformcited by 6
  6. 2023Tractability of approximation by general shallow networkscited by 4
  7. 2024Spherical Analysis of Learning Nonlinear Functionalscited by 4

Built on by

Not cited yet.