{
  "abstract": "Orthogonal and Stiefel constraints are widely used in neural network optimization to control weight geometry, improve conditioning, and stabilize signal propagation. Yet standard Stiefel optimization imposes a hidden restriction: while learning the directions of a linear map, it fixes every singular value to one, preventing the model from learning how strongly different directions should be amplified or suppressed. We introduce ManifoldFlow, a spectrum-learnable extension of Stiefel optimization that separates directional geometry from spectral scaling through $W=QS^{1/2}$, where $Q^\\top Q=I$ and $S\\succ0$. Beyond simply making the spectrum trainable, ManifoldFlow uses the gradient component rejected by the Stiefel constraint as a signal for how the spectrum should change, accumulates this signal across aligned tangent spaces, and jointly updates the orthonormal basis and a bounded SPD spectrum. This construction preserves exact spectral interpretability because $W^\\top W=S$, while recovering standard Stiefel optimization when $S=I$. We establish the geometry of the rejected gradient, characterize optimization over the resulting Stiefel--SPD product space, and derive how the learned spectrum affects conditioning, signal propagation, and spectral complexity. Across diverse tasks, ManifoldFlow consistently outperforms matched fixed-spectrum Stiefel baselines, with particularly strong gains in recurrent language modeling. Our code is available at https://github.com/Hik289/manifold_flow.git.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.04535",
  "author_total": 2,
  "authors": [
    {
      "name": "Haiwen Yi"
    },
    {
      "name": "Xinyuan Song"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/btn4ccdzmkm7z.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.04535",
  "fulltext_url": null,
  "id": "btn4ccdzmkm7z",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.04535",
  "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",
      "stat.ML"
    ],
    [
      "openalex",
      "1702|Neural Networks and Applications"
    ]
  ],
  "openalex": "W7167599673",
  "pdf_url": "https://arxiv.org/pdf/2607.04535",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/btn4ccdzmkm7z.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:00Z",
      "native_id": "2607.04535",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.04535"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:stat.ML",
      "subfield": "statistics"
    }
  ],
  "title": "ManifoldFlow: Spectrum-Learnable Stiefel Optimization via SPD Relaxation",
  "title_text": "ManifoldFlow: Spectrum-Learnable Stiefel Optimization via SPD Relaxation",
  "type": "preprint",
  "updated_at": "2026-10-30T12:30:00Z",
  "url": "https://research.draconitic.com/paper/btn4ccdzmkm7z",
  "venue": "arXiv",
  "versions": []
}