{
  "abstract": "We develop inference for manifold-valued population minimizers when each observation belongs to a different participant and only locally private messages reach the analyst. The method releases randomized tangent gradients and combines them through Riemannian stochastic approximation and Polyak-Ruppert averaging. Directly inserting a private data surrogate into a nonlinear loss can shift its population target, whereas conditional centring of the released gradient preserves the first-order equation. We introduce symmetric-pair regression (SPR) to estimate the asymptotic variance from the same private messages used for point estimation, without holding out participants or requesting a second release. We prove the central limit theorem and consistency of the fully transcript-based sandwich covariance and intrinsic Wald region under local differential privacy. Simulations across various statistical problems and manifolds support the predicted decrease in estimation error and near-nominal coverage under moderate privacy. An application to NHANES anthropometric data illustrates private estimation of a leading body-size direction and its uncertainty.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.22642",
  "author_total": 3,
  "authors": [
    {
      "name": "Xiaotian Chang"
    },
    {
      "name": "Yangdi Jiang"
    },
    {
      "name": "Qirui Hu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1d0ve1tpwh30r.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.22642",
  "fulltext_url": null,
  "id": "1d0ve1tpwh30r",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.22642",
  "language": null,
  "licence": "CC-BY-SA-4.0",
  "licence_label": "CC BY-SA 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-sa/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.LG"
    ],
    [
      "arxiv",
      "math.OC"
    ],
    [
      "arxiv",
      "math.ST"
    ],
    [
      "arxiv",
      "stat.ML"
    ],
    [
      "arxiv",
      "stat.TH"
    ],
    [
      "openalex",
      "1702|Privacy-Preserving Technologies in Data"
    ]
  ],
  "openalex": "W7214053327",
  "pdf_url": "https://arxiv.org/pdf/2609.22642",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1d0ve1tpwh30r.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:00Z",
      "native_id": "2609.22642",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.22642"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.ST",
      "subfield": "statistics"
    }
  ],
  "title": "Locally Private Inference for Riemannian Stochastic Optimization",
  "title_text": "Locally Private Inference for Riemannian Stochastic Optimization",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:06Z",
  "url": "https://research.draconitic.com/paper/1d0ve1tpwh30r",
  "venue": "arXiv",
  "versions": []
}