{
  "abstract": "The performance of gradient methods depends critically on step-size selection. Spectral step-size design involves constructing candidates from finite history and ordering them during iteration. We address both through a finite-moment spectral framework. For candidate construction, we relate Huang--Dai--Liu determinant pencils, limited-memory steepest-descent Krylov--Ritz extraction, pseudo-memory moment realization, and Gu--Du recovery of spectral nodes and component weights. The framework distinguishes generalized-eigenvalue recoverability, finite-window implementability, and component-weight interpretability. Under explicit positivity, rank-one, regularity, and exact-recovery assumptions, these properties become compatible in a common finite-moment regime, with the long, short, and mean LMSD constructions as representatives. For ordering, we identify spectral rebound: a step targeting a lower spectral scale may reactivate previously reduced high-frequency components. Idealized one-target, block, and recomputed memory-\\(m\\) models yield phase-count recurrences motivating a high-frequency-first strategy with spectral refreshes. We propose a rebound-aware spectral gradient method (RASG) that ranks Ritz candidates by frequency and estimated strength, and sets phase duration via a Ritz-uncertainty estimator. Two variants, RASG-KT and RASG-LMSD, use a Kato--Temple-inspired estimator and a long/short LMSD discrepancy. For strictly convex quadratics, global \\(R\\)-linear convergence holds under an admissibility safeguard. Experiments on ill-conditioned quadratic problems and two large-scale convex problems illustrate the behavior.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2606.25311",
  "author_total": 3,
  "authors": [
    {
      "name": "Yijia Zhou"
    },
    {
      "name": "Ran Gu"
    },
    {
      "name": "Ya-xiang Yuan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/56179fnp0zj9m.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2606.25311",
  "fulltext_url": null,
  "id": "56179fnp0zj9m",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2606.25311",
  "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",
      "math.OC"
    ],
    [
      "openalex",
      "2612|Advanced Optimization Algorithms Research"
    ]
  ],
  "openalex": "W7165933453",
  "pdf_url": "https://arxiv.org/pdf/2606.25311",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-23",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/56179fnp0zj9m.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:00Z",
      "native_id": "2606.25311",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2606.25311"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Structured Spectral Step-Sizes and Rebound-Aware Ordering for Gradient Methods",
  "title_text": "Structured Spectral Step-Sizes and Rebound-Aware Ordering for Gradient Methods",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:00Z",
  "url": "https://research.draconitic.com/paper/56179fnp0zj9m",
  "venue": "arXiv",
  "versions": []
}