{
  "abstract": "We study kernel functions induced by a linear time-invariant (LTI) stochastic differential equation (SDE), an approach that embeds prior knowledge about the function to be estimated directly into the state-space description of the SDE. This kernel-design perspective is relevant across the many settings where kernels play a central role, including kernel methods in statistics, Gaussian process (GP) regression in machine learning, and kernel-based regularization methods in system identification. The resulting class of SDE-induced kernels includes several kernels of practical interest, such as Matérn kernels with half-integer smoothness, spline kernels, and related kernels used in system identification. In contrast to existing linear-time GP regression methods for these kernels, which are typically based on Kalman filtering and Rauch--Tung--Striebel smoothing on the latent state, we work directly with the SSS representation of the observed covariance matrix. This yields linear-complexity recursive algorithms for the matrix operations required in GP regression, including factorization, linear solves, log determinants, posterior covariance, log-marginal-likelihood gradients, and posterior sampling; the same generator-level recursions extend, with only minor modifications, to the analogous computations in kernel-based regularization methods for system identification. Compared with previous uses of SSS representations in GP regression, we derive efficient recursive algorithms for analytic marginal-likelihood gradients in three application-relevant cases, and we develop new recursions for posterior computations and Matheron-type sampling. Numerical experiments verify the accuracy of the proposed recursions and illustrate posterior inference and sampling for representative SDE-induced kernels.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.05856",
  "author_total": 5,
  "authors": [
    {
      "name": "Mikkel Paltorp"
    },
    {
      "name": "Luigi Caglio"
    },
    {
      "name": "Tianshi Chen"
    },
    {
      "name": "Evangelos Katsanos"
    },
    {
      "name": "Martin S. Andersen"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/b93vyf6kaf9nt.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.05856",
  "fulltext_url": null,
  "id": "b93vyf6kaf9nt",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.05856",
  "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",
      "cs.NA"
    ],
    [
      "arxiv",
      "math.NA"
    ],
    [
      "arxiv",
      "stat.ML"
    ],
    [
      "openalex",
      "1702|Gaussian Processes and Bayesian Inference"
    ]
  ],
  "openalex": "W7220661235",
  "pdf_url": "https://arxiv.org/pdf/2610.05856",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/b93vyf6kaf9nt.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:10Z",
      "native_id": "2610.05856",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.05856"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:stat.ML",
      "subfield": "statistics"
    }
  ],
  "title": "On the Sequentially Semiseparable Structure of SDE-Induced Kernels and Efficient Algorithms for Kernel-Based Estimation Problems",
  "title_text": "On the Sequentially Semiseparable Structure of SDE-Induced Kernels and Efficient Algorithms for Kernel-Based Estimation Problems",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:16Z",
  "url": "https://research.draconitic.com/paper/b93vyf6kaf9nt",
  "venue": "arXiv",
  "versions": []
}