{
  "abstract": "We derive a fast factorization of the exact Karhunen--Loève transform (KLT) of an AR(1) source by mapping it onto the Discrete Cosine Transform. For even $N$ and every $ρ\\in(0,1)$, the KLT factors exactly into Chen's fast DCT-II structure---butterfly plus fixed half-size DCT-II and DCT-IV cores---completed by two orthogonal corrections: eigenvector matrices of diagonal-plus-rank-one matrices carrying the entire $ρ$-dependence. Fast DCT factorizations are reused unchanged; as $ρ\\to1$ the corrections become identities, recovering Chen's algorithm. At short lengths the design is explicit: at $N=4$ the transform is one butterfly and two rotations with $\\tan2ψ_s=-2/(1-ρ)$, $\\tan2ψ_a=2/(1+ρ)$; at $N=8$ every correction entry is in radicals, via one quartic serving both branches, and factors into six Givens rotations. Complete and scaled realizations run the exact 8-point KLT in 32 and 24 multiplications, about twice the fixed DCT-II; at $N=4$, a single extra multiplication. At large $N$, fast-multipole application of the corrections yields the exact KLT in $O(N\\log N)$ operations to prescribed accuracy; this overhead thus peaks at intermediate sizes and vanishes in both limits.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.20221",
  "author_total": 1,
  "authors": [
    {
      "name": "Yuriy A. Reznik"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/38xxsrwjn3y9d.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.20221",
  "fulltext_url": null,
  "id": "38xxsrwjn3y9d",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.20221",
  "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",
      "cs.IT"
    ],
    [
      "arxiv",
      "cs.NA"
    ],
    [
      "arxiv",
      "math.IT"
    ],
    [
      "arxiv",
      "math.NA"
    ]
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  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2609.20221",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-18",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/38xxsrwjn3y9d.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:07Z",
      "native_id": "2609.20221",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.20221"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "Exact fast factorizations of the AR(1) Karhunen-Loeve transform",
  "title_text": "Exact fast factorizations of the AR(1) Karhunen-Loeve transform",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:07Z",
  "url": "https://research.draconitic.com/paper/38xxsrwjn3y9d",
  "venue": "arXiv",
  "versions": []
}