{
  "abstract": "We study principal component analysis (PCA) under memory constraints, a setting that is increasingly important in large-scale data analysis. Our focus is on Oja's algorithm, which is a one-pass, memory-efficient algorithm requiring only $O(p)$ storage in the rank-one case and $O(pk)$ storage for $k$- PCA. The main goal is to develop a procedure based on Oja's algorithm that is optimal without prior knowledge of the eigengap, which is otherwise needed to tune the learning rates effectively. We do this by first introducing a geometric perspective on the convergence of $k$-PCA: by embedding the Oja iterates into the exterior space, we establish an exact equivalence between $k$-PCA in the ambient space and $1$-PCA in the exterior space. This geometric viewpoint enables us to establish several convergence guarantees, including one based only on a single learning-rate schedule. Building on this geometric perspective, we develop an averaging theory for $k$-PCA and show that the resulting averaged estimator is adaptive: it achieves the nearly optimal convergence rate without prior knowledge of the eigengap. As an application, we construct memory-efficient estimators for elliptical component analysis (ECA) \\cite{han2014scale,han2018eca}. Simulation studies and real data analysis are conducted to demonstrate the benefits of our proposed algorithms.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.23332",
  "author_total": 3,
  "authors": [
    {
      "name": "Tuan Pham"
    },
    {
      "name": "Alessandro Rinaldo"
    },
    {
      "name": "Purnamrita Sarkar"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0wmthdm5t0m63.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.23332",
  "fulltext_url": null,
  "id": "0wmthdm5t0m63",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.23332",
  "language": null,
  "licence": "CC0-1.0",
  "licence_label": "CC0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/publicdomain/zero/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.ST"
    ],
    [
      "arxiv",
      "stat.ME"
    ],
    [
      "arxiv",
      "stat.TH"
    ],
    [
      "openalex",
      "2613|Statistical Methods and Applications"
    ]
  ],
  "openalex": "W7214009640",
  "pdf_url": "https://arxiv.org/pdf/2609.23332",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-12",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0wmthdm5t0m63.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2609.23332",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.23332"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.ST",
      "subfield": "statistics"
    }
  ],
  "title": "Streaming PCA: averaging from a geometric perspective",
  "title_text": "Streaming PCA: averaging from a geometric perspective",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:26Z",
  "url": "https://research.draconitic.com/paper/0wmthdm5t0m63",
  "venue": "arXiv",
  "versions": []
}