{
  "abstract": "Streaming principal component analysis (PCA) seeks to recover a leading spectral subspace in a single pass over a data stream. We give a new analysis of the ubiquitous Oja's algorithm [Oja82] for the most general, gap-free variant of this problem, where no eigengap assumptions are made on the underlying mean matrix, complemented by a nearly-matching lower bound. Prior works achieving near-optimal rates for streaming PCA either required gap assumptions [JJK+16, HNWW21], or were limited to rank-one updates [AZL17, Lia23]. Our proof only uses a second moment bound on the individual stochastic updates, bypassing the almost sure bounds needed by prior near-optimal analyses, and the analogous offline matrix Bernstein bound. We also extend our result to a Rayleigh quotient notion of approximate PCA, addressing an open question of [JJK+16]. As our main application, we give gap-free differentially private PCA guarantees for sub-Gaussian data, settling Conjecture 1.1 of [Bro26] up to logarithmic factors.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.26508",
  "author_total": 4,
  "authors": [
    {
      "name": "Anming Gu"
    },
    {
      "name": "Syamantak Kumar"
    },
    {
      "name": "Kevin Tian"
    },
    {
      "name": "Chutong Yang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bf6fs3mavsakq.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.26508",
  "fulltext_url": null,
  "id": "bf6fs3mavsakq",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.26508",
  "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.DS"
    ],
    [
      "arxiv",
      "cs.LG"
    ],
    [
      "arxiv",
      "stat.ML"
    ],
    [
      "openalex",
      "1702|Stochastic Gradient Optimization Techniques"
    ]
  ],
  "openalex": "W7214065390",
  "pdf_url": "https://arxiv.org/pdf/2609.26508",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bf6fs3mavsakq.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:49Z",
      "native_id": "2609.26508",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.26508"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:stat.ML",
      "subfield": "statistics"
    }
  ],
  "title": "Gap-Free Streaming PCA Beyond Rank-One Updates: Near-Optimal Rates and Applications to Differential Privacy",
  "title_text": "Gap-Free Streaming PCA Beyond Rank-One Updates: Near-Optimal Rates and Applications to Differential Privacy",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:49Z",
  "url": "https://research.draconitic.com/paper/bf6fs3mavsakq",
  "venue": "arXiv",
  "versions": []
}