{
  "abstract": "We derive the complete duality theory underlying the Hodrick--Prescott filter, whose rank-deficient second-difference penalty admits infinitely many equivalent trend representations via generalized inverses. Constructing two canonical choices---the Moore--Penrose-based \\emph{B-representation} and an alternative \\emph{A-representation}---we prove that the extracted trend is invariant across representations, obtain a closed-form Bregman-type divergence quantifying their disagreement under a shared coefficient vector, and show this divergence vanishes as $λ\\to\\infty$. We further mollify the non-smooth $\\ell_1$ trend filter with a compactly supported biweight kernel to obtain a closed-form $C^3$ \\emph{Convolutional LASSO} that restores Newton-type quadratic convergence without sacrificing the kink- setecting character of $\\ell_1$ regularization. Theoretical results are verified numerically and benchmarked against ADMM/IRLS on NVIDIA's 2013--2018 daily closing prices, where the sparse filter isolates genuine growth-regime breaks---chiefly the November 2016 post-earnings acceleration---cleanly separated from the smooth $L_2$ trend.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.29821",
  "author_total": 1,
  "authors": [
    {
      "name": "Shintaro Yoshizawa"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/e6qx9snxswsy1.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.29821",
  "fulltext_url": null,
  "id": "e6qx9snxswsy1",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.29821",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "stat.ME"
    ],
    [
      "openalex",
      "2613|Statistical Methods and Inference"
    ]
  ],
  "openalex": "W7204921282",
  "pdf_url": "https://arxiv.org/pdf/2608.29821",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/e6qx9snxswsy1.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:54Z",
      "native_id": "2608.29821",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.29821"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:stat.ME",
      "subfield": "statistics"
    }
  ],
  "title": "A Generalized Ridge Regression and Convolutional LASSO",
  "title_text": "A Generalized Ridge Regression and Convolutional LASSO",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:54Z",
  "url": "https://research.draconitic.com/paper/e6qx9snxswsy1",
  "venue": "arXiv",
  "versions": []
}