{
  "abstract": "We study an oracle subspace-perturbation model for 3D localization. The procedure is given an $s$-dimensional subspace $\\widetilde{\\mathcal U}$ and a deterministic error bound \\[ \\varepsilon_{\\mathrm{sub}} = \\left\\|P_{\\widetilde{\\mathcal U}}-P_{\\mathcal U}\\right\\|_2, \\] where $\\mathcal{U}$ is an $s$-dimensional subspace of far-field patterns with wavenumber $κ$, and $P_{\\mathcal{V}}$ denotes orthogonal projection onto a subspace $\\mathcal{V}$. Under explicit separation and conditioning hypotheses for arbitrary point clouds, we prove that the perturbed spherical MUSIC objective \\[ \\widetilde{q}({\\mathbf z}) = 1-\\left\\|P_{\\widetilde{\\mathcal{U}}}φ_{\\mathbf z}\\right\\|_2^2 \\] has a unique strongly convex well in every ball $B_{γ/κ}(x_j)$ and a uniform value gap outside the union of the certified wells. A fixed-step gradient map with $h\\asympκ^{-2}$ leaves each well invariant and converges linearly to its unique minimizer. Consequently, thresholding on an $O(κ^{-1})$-mesh, followed by gradient descent and duplicate removal, recovers all relevant minima with localization bound \\[ \\mathfrak{R}(\\varepsilon_{\\mathrm{sub}}) \\lesssim \\frac{\\varepsilon_{\\mathrm{sub}}}κ. \\] The arbitrary-cloud frame analysis yields the sufficient condition \\[ κδ_X\\gtrsim s^{2/3} \\] through an absolute coherence row sum and Gershgorin's theorem. We construct lower-frame counterexamples below the $s^{1/6}$ scale, upper-frame counterexamples below the $s^{1/3}$ scale, and examples showing that $2/3$ is optimal for the absolute-row-sum argument. Finally, for parameter classes containing a uniformly admissible one-point displacement path, we prove \\[ \\mathfrak{R}(\\varepsilon_{\\mathrm{sub}}) \\sim \\frac{\\varepsilon_{\\mathrm{sub}}}κ. \\]",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.03264",
  "author_total": 2,
  "authors": [
    {
      "name": "Albert Fannjiang"
    },
    {
      "name": "Chuong Nguyen"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ed7efdsc3bngy.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.03264",
  "fulltext_url": null,
  "id": "ed7efdsc3bngy",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.03264",
  "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",
      "1711|Direction-of-Arrival Estimation Techniques"
    ]
  ],
  "openalex": "W7208838800",
  "pdf_url": "https://arxiv.org/pdf/2609.03264",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ed7efdsc3bngy.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:29Z",
      "native_id": "2609.03264",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.03264"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Certified Spherical MUSIC for 3D Localization under Adversarial Subspace Perturbations",
  "title_text": "Certified Spherical MUSIC for 3D Localization under Adversarial Subspace Perturbations",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/ed7efdsc3bngy",
  "venue": "arXiv",
  "versions": []
}