{
  "abstract": "Sparse Bayesian learning is widely used for sparse linear inverse problems, yet its large-system stationary behavior remains poorly understood because all variance hyperparameters are estimated from the same data. We study classical sparse Bayesian learning, formulated as evidence maximization (type-II maximum likelihood), for underdetermined linear models with sensing matrices having independent and identically distributed Gaussian entries, Gaussian measurement noise, and an unknown deterministic signal sequence. Hyperparameter reoptimization induces a nonvanishing feedback term: a typical coordinate obeys a reoptimization-corrected scalar Gaussian law whose signal coefficient is governed by the normalized adaptive response rather than the frozen resolvent trace. A one-coordinate leave-one-out construction gives an exact conditional Gaussian law, which is transferred to a selected full stationary branch without assuming asymptotic closeness of the reduced and full stationary vectors. Combining this law with the Karush--Kuhn--Tucker conditions of the evidence objective yields a generally set-valued scalar relation and three branchwise large-system consistency relations. If the model noise variance is jointly estimated by evidence maximization, interior joint stationarity yields an exact finite-dimensional equality between the normalized residual energy and normalized resolvent trace. When the limiting signal law has nonzero mass at zero, this identity further yields a parameter-free asymptotic chi-square null law. Under an additional differentiability condition on the selected scalar branch, the large-system characterization also gives a closed relation for the reconstruction error of the posterior mean. The analysis is stationary-point based and permits multiple stationary branches.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.06560",
  "author_total": 2,
  "authors": [
    {
      "name": "Fangqing Xiao"
    },
    {
      "name": "Dirk T. M. Slock"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cp6tr1e0y4hyz.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.06560",
  "fulltext_url": null,
  "id": "cp6tr1e0y4hyz",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.06560",
  "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.IT"
    ],
    [
      "arxiv",
      "eess.SP"
    ],
    [
      "arxiv",
      "math.IT"
    ],
    [
      "openalex",
      "2206|Sparse and Compressive Sensing Techniques"
    ]
  ],
  "openalex": "W7212008566",
  "pdf_url": "https://arxiv.org/pdf/2609.06560",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-06",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cp6tr1e0y4hyz.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2609.06560",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.06560"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.IT",
      "subfield": "general"
    }
  ],
  "title": "Large-System Analysis of Sparse Bayesian Learning",
  "title_text": "Large-System Analysis of Sparse Bayesian Learning",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/cp6tr1e0y4hyz",
  "venue": "arXiv",
  "versions": []
}