{
  "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 $\\|\\mathrm{L} f\\|^2_{L_2}$, where $\\mathrm{L} $ is a differential-like (spline-admissible) operator. The natural domain of the minimization is the native space $\\mathcal{H}_\\mathrm{L}$ 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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.14920",
  "author_total": 1,
  "authors": [
    {
      "name": "Michael Unser"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/eanj4pytv2p49.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.14920",
  "fulltext_url": null,
  "id": "eanj4pytv2p49",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.14920",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.FA"
    ],
    [
      "openalex",
      "2610|Numerical methods in inverse problems"
    ]
  ],
  "openalex": "W7221805309",
  "pdf_url": "https://arxiv.org/pdf/2610.14920",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-10",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/eanj4pytv2p49.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2610.14920",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.14920"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Splines and Operators: Part II - Resolution of Inverse Problems",
  "title_text": "Splines and Operators: Part II - Resolution of Inverse Problems",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/eanj4pytv2p49",
  "venue": "arXiv",
  "versions": []
}