Splines and Operators: Part II - Resolution of Inverse Problems
Abstract
We present a unified treatment of the data-fitting problems with quadratic (Hilbertian) regularization that arise in approximation theory, machine learning, engineering, and statistics. The generic problem is an optimization in the continuum: given M linear measurements of an unknown function, one minimizes a convex data-fidelity term that is penalized by |L f|2L2, where $\mathrm{L} $ is a differential-like (spline-admissible) operator. The natural domain of the minimization is the native space ℋL of $\mathrm{L} $ - the largest Hilbert space over which the regularizer remains meaningful - for which we provide a concise and directly usable characterization. Central to our formulation is a measurement-adapted (conditioned) basis that decouples the null-space and spline components of the solution. This basis serves in three places. First, it tightens the formulation of our representer theorem, which confines the unique solution to an M-dimensional space without any orthogonality constraint on the coefficients. Second, it streamlines the implementation, as the exact discretization of the problem becomes an unconstrained convex program with closed-form solutions in the least-squares case, which covers ridge regression, Tikhonov regularization, and smoothing splines. Third, it makes the bridge with the minimum-mean-square-error estimation of generalized Gaussian processes, which is the topic of a companion paper.
Related
- Splines and Operators: Part I - Admissibility and Native Spaces
- Inverse problems with fractional operators: an analytical study and its numerical linear algebra treatment via the GLT theory
- Optimization Inverse Spectral Problems for Sturm--Liouville Operators with Potentials in Lp(0,1)
- Posterior Convergence without Force Convergence: Resolution-Stable Sampling for Rough Bayesian Inverse Problems
- PH2T-splines, Part I: A Reasonable Mesh Assumption
BibTeX
@misc{unser2026splines,
title = {{Splines and Operators: Part II - Resolution of Inverse Problems}},
author = {Michael Unser},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.14920},
eprint = {2610.14920},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.14920},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Splines and Operators: Part II - Resolution of Inverse Problems
AU - Michael Unser
PY - 2026
DA - 2026/10/10
PB - arXiv
DO - 10.48550/arxiv.2610.14920
UR - https://arxiv.org/abs/2610.14920
AB - We present a unified treatment of the data-fitting problems with quadratic (Hilbertian) regularization that arise in approximation theory, machine learning, engineering, and statistics. The generic problem is an optimization in the continuum: given M linear measurements of an unknown function, one minimizes a convex data-fidelity term that is penalized by |L f|^2_L_2, where $\mathrm{L} $ is a differential-like (spline-admissible) operator. The natural domain of the minimization is the native space ℋ_L of $\mathrm{L} $ - the largest Hilbert space over which the regularizer remains meaningful - for which we provide a concise and directly usable characterization. Central to our formulation is a measurement-adapted (conditioned) basis that decouples the null-space and spline components of the solution. This basis serves in three places. First, it tightens the formulation of our representer theorem, which confines the unique solution to an M-dimensional space without any orthogonality constraint on the coefficients. Second, it streamlines the implementation, as the exact discretization of the problem becomes an unconstrained convex program with closed-form solutions in the least-squares case, which covers ridge regression, Tikhonov regularization, and smoothing splines. Third, it makes the bridge with the minimum-mean-square-error estimation of generalized Gaussian processes, which is the topic of a companion paper.
ER -