{
  "abstract": "A matrix cascade is an iterative refinement process in which vector-valued data are repeatedly filtered by matrix-valued masks across successively finer scales. Such cascades arise in multichannel subdivision, Hermite refinement, multiresolution synthesis, and exponential-spline constructions, and in nonstationary schemes the masks may vary with scale. We develop a chronological stability framework for this setting, allowing both the transition operators and the associated matching spaces to change from level to level. If the chronological radius satisfies $ρ_{\\rm ch}<1$ and the matching defects are summable, then the cascade converges uniformly and, for every $q\\in(ρ_{\\rm ch},1)$, \\[ \\|F_n-Φ\\|_\\infty \\lesssim q^n+\\sum_{j=1}^nδ_j q^{\\,n-j} +\\sum_{j>n}δ_j . \\] The time-weighted defect term records the level at which each perturbation is inserted, and the matching spaces need not converge; they may even alternate indefinitely. For compact piecewise-linear seeds, every finite ordered matrix cascade admits an exact fixed-width ReLU realization with $O(n)$ depth and parameters, and this linear depth order is worst-case optimal at fixed width. Consequently, exponentially decaying defects yield certified approximants of size $O(\\log\\varepsilon^{-1})$. We further construct stable two-channel classes in which the chronological radius and the leading metric-entropy coefficient are independent invariants, showing that stability does not determine finite-accuracy description complexity. The framework therefore provides tolerance-controlled finite representations for nonstationary multiresolution and exponential-spline refinement.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.13981",
  "author_total": 2,
  "authors": [
    {
      "name": "Yuwen Li"
    },
    {
      "name": "Yupeng Wang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5te2vge0nthbx.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.13981",
  "fulltext_url": null,
  "id": "5te2vge0nthbx",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.13981",
  "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.NA"
    ],
    [
      "arxiv",
      "math.NA"
    ],
    [
      "openalex",
      "2604|Mathematical Analysis and Transform Methods"
    ]
  ],
  "openalex": "W7213337162",
  "pdf_url": "https://arxiv.org/pdf/2609.13981",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-12",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5te2vge0nthbx.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:29Z",
      "native_id": "2609.13981",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.13981"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "Chronological stability for nonstationary matrix refinement",
  "title_text": "Chronological stability for nonstationary matrix refinement",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/5te2vge0nthbx",
  "venue": "arXiv",
  "versions": []
}