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Lp approximation results for infinite dimensional Neural Networks
Abstract
Leveraging the neural architectures which we introduced in arXiv:2109.13512v4, we show a global universal approximation theorem in the topology of Lp(μ), where 1≤ p<∞ and μ is a Radon probability measure on a suitable infinite dimensional topological space X. Namely, any function f:X→ ℝ in Lp(μ) can be approximated to any degree of accuracy by suitable infinite dimensional architectures. These architectures can be in turn approximated by almost classical neural networks which are specified by a finite number of parameters only. The vectorial case (where f=f(x)∈ E and E is a Banach space) is also considered and analogous results are obtained.
Related
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BibTeX
@misc{galimberti2026approximation,
title = {{$L^p$ approximation results for infinite dimensional Neural Networks}},
author = {Luca Galimberti},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.08230},
eprint = {2608.08230},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.08230},
}
RIS
TY - UNPB
TI - L^p approximation results for infinite dimensional Neural Networks
AU - Luca Galimberti
PY - 2026
DA - 2026/10/28
PB - arXiv
DO - 10.48550/arxiv.2608.08230
UR - https://arxiv.org/abs/2608.08230
AB - Leveraging the neural architectures which we introduced in arXiv:2109.13512v4, we show a global universal approximation theorem in the topology of L^p(μ), where 1≤ p<∞ and μ is a Radon probability measure on a suitable infinite dimensional topological space X. Namely, any function f:X→ ℝ in L^p(μ) can be approximated to any degree of accuracy by suitable infinite dimensional architectures. These architectures can be in turn approximated by almost classical neural networks which are specified by a finite number of parameters only. The vectorial case (where f=f(x)∈ E and E is a Banach space) is also considered and analogous results are obtained.
ER -