{
  "abstract": "The choice of Riemannian metric can strongly influence the convergence of gradient-based optimization over covariance matrices. Euclidean, Bures-Wasserstein and affine-invariant metrics are common choices, but their relative effectiveness depends on the objective. We introduce a two-parameter family defined by $X^{p}LX^{q}+X^{q}LX^{p}=U$, solved for $L$ at each tangent vector $U$, that contains all three as exact members, at $(0,0)$, $(1,0)$ and $(1,1)$, and extends past them. We treat the choice of member as a particular way of preconditioning for a given problem. To this end, we analyze the conditioning of the Riemannian Hessian at the solution. We show that it obeys a lower bound that depends on $(p,q)$ only through the exponent $r=p+q$. When the Euclidean Hessian is a pure power that mixes no eigendirections, the member $p=q=r/2$ attains that bound, and a closed-form criterion identifies the other members that do. We discuss ways to tune $r$ for a given problem. Experiments on real covariance data confirm the predicted conditioning and the benefit of tuning $r$. A task covariance example shows a further gain from tuning the shape.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.17089",
  "author_total": 4,
  "authors": [
    {
      "name": "Yibang Li"
    },
    {
      "name": "Bamdev Mishra"
    },
    {
      "name": "Pratik Jawanpuria"
    },
    {
      "name": "Cyrus Mostajeran"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1tyk2xyensmz0.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.17089",
  "fulltext_url": null,
  "id": "1tyk2xyensmz0",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.17089",
  "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.LG"
    ],
    [
      "arxiv",
      "math.OC"
    ],
    [
      "openalex",
      "2612|Advanced Optimization Algorithms Research"
    ]
  ],
  "openalex": "W7213430069",
  "pdf_url": "https://arxiv.org/pdf/2609.17089",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1tyk2xyensmz0.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:45Z",
      "native_id": "2609.17089",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.17089"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Optimization over covariance matrices with a parameterized metric",
  "title_text": "Optimization over covariance matrices with a parameterized metric",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:45Z",
  "url": "https://research.draconitic.com/paper/1tyk2xyensmz0",
  "venue": "arXiv",
  "versions": []
}