{
  "abstract": "Row-sparse projection provides a useful tool in machine learning (ML) when it comes to, for example, feature selection, aiming to choose most relevant features for various ML objectives. One way to seek a high quality row-sparse projection is to combine an ML objective, such as the ones for PCA, LDA, and OCCA, with the matrix $(2,1)$-norm regularization which is nonsmooth. Such combinations result in challenging optimization problems on the Stiefel manifold that need to be solved efficiently. In this paper, a unifying NEPv framework is established to efficiently deal with optimization on the Stiefel manifold with the $(2,1)$-norm regularization. The effect of the $(2,1)$-norm regularization is also investigated. The wide applicability of the framework is demonstrated through the combinations of common learning objectives in today's data science applications with the $(2,1)$-norm regularization. Numerical experiments are presented to illustrate the use of the NEPv approach and to gain insights as to what a proper regularizing parameter should have in real-world applications.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.26675",
  "author_total": 4,
  "authors": [
    {
      "name": "Ren-Cang Li"
    },
    {
      "name": "Li Wang"
    },
    {
      "name": "Lei-Hong Zhang"
    },
    {
      "name": "Zhaojun Bai"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fcwpsq0egwp3g.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.26675",
  "fulltext_url": null,
  "id": "fcwpsq0egwp3g",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.26675",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.NA"
    ],
    [
      "arxiv",
      "math.NA"
    ],
    [
      "openalex",
      "2206|Sparse and Compressive Sensing Techniques"
    ]
  ],
  "openalex": "W7214066432",
  "pdf_url": "https://arxiv.org/pdf/2609.26675",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fcwpsq0egwp3g.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:43Z",
      "native_id": "2609.26675",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.26675"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "NEPv Approach for Optimization on Stiefel Manifold with the $(2,1)$-norm Regularization",
  "title_text": "NEPv Approach for Optimization on Stiefel Manifold with the (2,1)-norm Regularization",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:43Z",
  "url": "https://research.draconitic.com/paper/fcwpsq0egwp3g",
  "venue": "arXiv",
  "versions": []
}