{
  "abstract": "In this second paper on descent methods as proof mechanisms in harmonic analysis (the first treated Bourgain's $Λ(p)$ selection theorem), entropic mirror descent gives an improved restricted isometry bound, improving sampling and recovery bounds in the Fourier Ratio program. Let $H$ be a complex $n\\times n$ matrix with $H^*H=nI$ and $|H_{ij}|=1$. For fixed $0<\\varepsilon<\\frac14$ and $A_0>0$, independent Bernoulli row selection with expected cardinality $C_{\\varepsilon,A_0}r\\log\\frac{2en}{r}\\log(2r)\\leq m\\leq n/2$ preserves, after normalization by $\\sqrt m$, the squared norm of every complex vector $y$ with $\\|y\\|_1\\leq\\sqrt r\\|y\\|_2$ within a factor $1\\pm\\varepsilon$, with failure probability at most $Cn^{-A_0}$. This includes every $r$-sparse vector and also fully supported ones; support size enters only through this inequality. For $y=\\widehat f$ the condition is exactly $\\operatorname{FR}(f)^2\\leq r$, where $\\operatorname{FR}(f)=\\|\\widehat f\\|_1/\\|\\widehat f\\|_2$, giving uniform energy sampling and approximate recovery even for signals with full Fourier support. The same holds for a uniform subset of prescribed cardinality, and for discrete Fourier and real Hadamard matrices. At fixed accuracy the bound removes one sparsity logarithm from the earlier count and replaces $\\log n$ by $\\log\\frac{2en}{r}$; for Walsh matrices it matches, up to constants, the lower bound of Błasiok et al.\\ in their range. One relative-entropy potential controls the corrections of an amplitude predictor at every scale; counting them on the symmetric difference of two samples gives the uniform estimate. We also prove the sufficient bound $C_{A_0}\\varepsilon^{-5}r\\log\\frac{2en}{r}\\log\\frac{2r}{\\varepsilon}$, and stable sparse recovery from $Cs\\log\\frac{2en}{s}\\log(2s)$ measurements, with error controlled by the noise and the best $s$-term approximation.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.22568",
  "author_total": 3,
  "authors": [
    {
      "name": "Will Burstein"
    },
    {
      "name": "Alex Iosevich"
    },
    {
      "name": "Ben Krause"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/34qymt65d0hvb.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.22568",
  "fulltext_url": null,
  "id": "34qymt65d0hvb",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.22568",
  "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",
      "math.CA"
    ],
    [
      "arxiv",
      "math.IT"
    ],
    [
      "openalex",
      "2206|Sparse and Compressive Sensing Techniques"
    ]
  ],
  "openalex": "W7214053819",
  "pdf_url": "https://arxiv.org/pdf/2609.22568",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-23",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/34qymt65d0hvb.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:40Z",
      "native_id": "2609.22568",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.22568"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Fourier Ratio and Restricted Isometry via Entropic Descent",
  "title_text": "Fourier Ratio and Restricted Isometry via Entropic Descent",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:40Z",
  "url": "https://research.draconitic.com/paper/34qymt65d0hvb",
  "venue": "arXiv",
  "versions": []
}