{
  "abstract": "The elementwise Hadamard product of two low-rank matrices provides a parameter-efficient model for data with multiplicative structure, but its modeling is challenging due to the presence of additional symmetries under coupled row/column scalings between the two factors. In order to leverage the geometry of the space, we formulate the learning of such matrices as optimization on a Riemannian quotient manifold. We propose a novel block-diagonal Riemannian metric derived from the pullback of the Frobenius inner product. The metric is shown to be invariant under these symmetries. We develop a Riemannian gradient descent algorithm that uses a tuning-free Gauss--Newton step size and scales linearly in the number of observed entries per iteration. The versatile framework of Riemannian quotient optimization enables both first-order and second-order Riemannian methods, the latter through a closed-form connection and the Riemannian Hessian. Experiments on real and synthetic datasets illustrate the efficacy of our proposed Riemannian approach.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2606.01216",
  "author_total": 3,
  "authors": [
    {
      "name": "Pratik Jawanpuria"
    },
    {
      "name": "Ankish Chandresh"
    },
    {
      "name": "Bamdev Mishra"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1v0zs74hwrg3b.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2606.01216",
  "fulltext_url": null,
  "id": "1v0zs74hwrg3b",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2606.01216",
  "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.LG"
    ],
    [
      "arxiv",
      "math.OC"
    ],
    [
      "openalex",
      "2206|Sparse and Compressive Sensing Techniques"
    ]
  ],
  "openalex": "W7163212004",
  "pdf_url": "https://arxiv.org/pdf/2606.01216",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-31",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1v0zs74hwrg3b.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:48Z",
      "native_id": "2606.01216",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2606.01216"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Riemannian Optimization for Hadamard Products of Low-Rank Matrices",
  "title_text": "Riemannian Optimization for Hadamard Products of Low-Rank Matrices",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:48Z",
  "url": "https://research.draconitic.com/paper/1v0zs74hwrg3b",
  "venue": "arXiv",
  "versions": []
}