Schauder Bases for $c[0, 1]$ Using ReLU, Softplus and Two Sigmoidal Functions
Abstract
We construct four Schauder bases for the space $C[0,1]$, one using ReLU functions, another using Softplus functions, and two more using sigmoidal versions of the ReLU and Softplus functions. This establishes the existence of a basis using these functions for the first time, and improves on the universal approximation property associated with them. We also show an $O(\frac{1}{n})$ approximation bound based on our ReLU basis, and a negative result on constructing multivariate functions using finite combinations of ReLU functions.
Cite
Text
Ganesh et al. "Schauder Bases for $c[0, 1]$ Using ReLU, Softplus and Two Sigmoidal Functions." Transactions on Machine Learning Research, 2025.Markdown
[Ganesh et al. "Schauder Bases for $c[0, 1]$ Using ReLU, Softplus and Two Sigmoidal Functions." Transactions on Machine Learning Research, 2025.](https://mlanthology.org/tmlr/2025/ganesh2025tmlr-schauder/)BibTeX
@article{ganesh2025tmlr-schauder,
title = {{Schauder Bases for $c[0, 1]$ Using ReLU, Softplus and Two Sigmoidal Functions}},
author = {Ganesh, Anand and Bose, Babhrubahan and Rajagopalan, Anand},
journal = {Transactions on Machine Learning Research},
year = {2025},
url = {https://mlanthology.org/tmlr/2025/ganesh2025tmlr-schauder/}
}