{
  "abstract": "Sparsity is a powerful structural resource in optimization and statistics. We develop frameworks for leveraging sparsity in sampling problems over the Hamming slice $\\mathcal{X}_k^d:=\\{\\mathbf{x}\\in\\{\\pm 1\\}^d:|\\{i:\\mathbf{x}_i=1\\}|=k\\}$, in high-dimensional regimes where $k\\ll d$ (i.e., where $\\mathcal{X}_k^d$ is \\emph{highly magnetized}). We use our frameworks to design improved samplers for canonical problems in the study of \\emph{Ising models} and \\emph{Bayesian sparse linear regression}. Our first main result considers the \\emph{Sherrington--Kirkpatrick} (SK) model restricted to fixed-magnetization slices $\\mathcal{X}_k^d$. We give a polynomial-time sampler for fixed-magnetization SK models at any inverse temperature $β>0$, under arbitrary external fields, provided that $k\\le c_βd$ for an appropriate constant $c_β$. By combining this result with an annealing strategy for estimating normalizing constants, we obtain polynomial-time samplers for the SK model at arbitrarily low temperatures under a sufficiently strong external field of strength $h$. In the large-$β$ limit, our framework permits sampling at field strengths within constant factors of the \\emph{Almeida--Thouless line} delineating the replica-symmetric and replica-symmetry-breaking regions ([dAT78]), improving polynomially over the field strength $h(β)$ required by the recent work of [BAR26]. Our second main result concerns the measurement complexity of polynomial-time Bayesian sparse linear regression. Recent work by [KSTZ25] shows how to sample from the canonical \\emph{Gaussian spike-and-slab posterior} with expected sparsity $k$, at any signal-to-noise ratio, given $n\\gtrsim k^3\\log^3 d$ Gaussian measurements. We improve this requirement to $n\\gtrsim k^{3/2}\\log^2 d+k\\log^3 d$, using a common sparsity-aware framework underlying both our results.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.08873",
  "author_total": 4,
  "authors": [
    {
      "name": "Syamantak Kumar"
    },
    {
      "name": "Purnamrita Sarkar"
    },
    {
      "name": "Kevin Tian"
    },
    {
      "name": "Yusong Zhu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0g17j0tv2xjr6.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.08873",
  "fulltext_url": null,
  "id": "0g17j0tv2xjr6",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.08873",
  "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.DS"
    ],
    [
      "arxiv",
      "cs.LG"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "arxiv",
      "math.ST"
    ],
    [
      "arxiv",
      "stat.ML"
    ],
    [
      "arxiv",
      "stat.TH"
    ],
    [
      "openalex",
      "2613|Markov Chains and Monte Carlo Methods"
    ]
  ],
  "openalex": "W7212082434",
  "pdf_url": "https://arxiv.org/pdf/2609.08873",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0g17j0tv2xjr6.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:01Z",
      "native_id": "2609.08873",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.08873"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.ST",
      "subfield": "statistics"
    }
  ],
  "title": "High-Magnetization Sampling at Low Temperatures: Ising Models and Bayesian Sparse Linear Regression",
  "title_text": "High-Magnetization Sampling at Low Temperatures: Ising Models and Bayesian Sparse Linear Regression",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:07Z",
  "url": "https://research.draconitic.com/paper/0g17j0tv2xjr6",
  "venue": "arXiv",
  "versions": []
}