A Finite Data Based Regularization in a Normed Linear Space Setting
Abstract
One of the basic problems in mathematical learning theory is to identify a function f: Ω→ ℝ with certain specific properties which fits a given training data {(xi, ξi)∈ Ω× ℝ: i=1, …, n}, in the sense that, f(xi) = ξi for i=1, …, n, where Ω is a compact subset of ℝd for some d∈ ℕ and f is required to have certain specified characteristics. We address this problem when f belongs to an arbitrary normed linear space X, and the evaluation maps f↦ f(xi) are replaced by maps of the form $ f\mapsto φ_i(f)$ on X, where φ1, …, φn are continuous linear functionals on X. Using an inner product structure on ℝn, we shall device a method of least-squares for obtaining an approximate solution for the above problem and also identify a subspace of X in which the least-square solution is unique, which in turn is shown to be equivalent to solving a matrix equation. In the context when the matrix under consideration is ill-conditioned, a regularized equation is devised, and order optimal error estimates are derived for the exact targeted data ξ=(ξ1, …, ξn) and also when it is noisy, by choosing the regularization parameter appropriately.
Related
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BibTeX
@misc{nair2026finite,
title = {{A Finite Data Based Regularization in a Normed Linear Space Setting}},
author = {M. Thamban Nair},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.11960},
eprint = {2609.11960},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.11960},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A Finite Data Based Regularization in a Normed Linear Space Setting
AU - M. Thamban Nair
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.11960
UR - https://arxiv.org/abs/2609.11960
AB - One of the basic problems in mathematical learning theory is to identify a function f: Ω→ ℝ with certain specific properties which fits a given training data {(x_i, ξ_i)∈ Ω× ℝ: i=1, …, n}, in the sense that, f(x_i) = ξ_i for i=1, …, n, where Ω is a compact subset of ℝ^d for some d∈ ℕ and f is required to have certain specified characteristics. We address this problem when f belongs to an arbitrary normed linear space X, and the evaluation maps f↦ f(x_i) are replaced by maps of the form $ f\mapsto φ_i(f)$ on X, where φ_1, …, φ_n are continuous linear functionals on X. Using an inner product structure on ℝ^n, we shall device a method of least-squares for obtaining an approximate solution for the above problem and also identify a subspace of X in which the least-square solution is unique, which in turn is shown to be equivalent to solving a matrix equation. In the context when the matrix under consideration is ill-conditioned, a regularized equation is devised, and order optimal error estimates are derived for the exact targeted data ξ=(ξ_1, …, ξ_n) and also when it is noisy, by choosing the regularization parameter appropriately.
ER -