arXiv:2609. 19937v1 Announce Type: cross Abstract: Recent studies have shown that smooth functions can be well approximated by ReLU neural networks with path norm constraint on the weights.
By Xianjun Li, Yunfei Yang
Recent studies have shown that smooth functions can be well approximated by ReLU neural networks with path norm constraint on the weights. We extend these results from uniform approximation to approximation in Sobolev norm.
This paper investigates the ρ^p-Lipschitz constants of deep ReLU neural networks with random weights drawn from a He‑style initialization. For zero‑bias networks, it provides high‑probability upper and lower bounds that differ by at most a logarithmic factor in depth, and shows a sharp contrast between the regimes p∈[1,2) and p∈[2,∞], with the former behaving like the Euclidean norm of a Gaussian vector and the latter like its dual norm. The analysis is extended to networks with non‑zero biases from symmetric distributions, yielding bounds that differ by a logarithmic factor in width and a linear factor in depth.
By Sjoerd Dirksen, Patrick Finke, Paul Geuchen, Dominik St\"oger, Felix Voigtlaender
arXiv:2609. 05263v1 Announce Type: cross Abstract: We investigate the best $L_2$ approximation of mixed Sobolev spaces by shallow neural networks with $n$ neurons and general activation functions.
By Yuwen Li, Guozhi Zhang
arXiv:2609. 03129v1 Announce Type: cross Abstract: Several classical machine-learning methods, such as KRRs and SVRs, are both computationally and analytically tractable since their estimators either admit closed-form expressions or are obtained by minimizing convex training objectives; neither feature is generally available for deep neural networks.
By Ruiyang Hong, Hrad Ghoukasian, Anastasis Kratsios
arXiv:2609.39020v1 Announce Type: new
Abstract: We study how well deep neural networks approximate and learn spectral Barron functions. Recent studies have shown that these function classes can be ef...
By Songqiu Ma, Yunfei Yang