{
  "abstract": "Conventional statistical modeling typically assumes a one-to-one mapping between input and output variables, where one-to-one is used in the predictive sense that a specified input results in a unique output. Many scientific and engineering systems, however, exhibit non-one-to-one(noto) input-output relations. In a noto relation, the same input may correspond to multiple admissible outputs, so the conventional statistical models may not be appropriate. This paper develops manifold regression, a parametric modeling framework for such noto input-output relations, which represents the underlying input-output relation through a latent manifold. The manifold is specified upto an unknown finite dimensional parameter and we estimate the unknown parameters from ordinary input-output observations by regularized profile optimization with a robust solver. Once the manifold has been learned, prediction is obtained by slicing the estimated manifold along a specified coordinate value. Because a slice may contain multiple latent roots, the resulting manifold prediction is naturally set-valued. This formulation extends predictive learning beyond conventional one-to-one statistical models while containing classical regression and inverse prediction as special cases. Theoretical results are established to justify the manifold regression model and its sliced prediction sets; and case studies are used to demonstrate stable performance across representative noto cases.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.11828",
  "author_total": 2,
  "authors": [
    {
      "name": "Yuanyuan Lin"
    },
    {
      "name": "Dennis K. J. Lin"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1dt5jv5hqxfcq.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.11828",
  "fulltext_url": null,
  "id": "1dt5jv5hqxfcq",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.11828",
  "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",
      "stat.ME"
    ],
    [
      "openalex",
      "2613|Advanced Statistical Methods and Models"
    ]
  ],
  "openalex": "W7221108082",
  "pdf_url": "https://arxiv.org/pdf/2610.11828",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1dt5jv5hqxfcq.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:28:03Z",
      "native_id": "2610.11828",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.11828"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:stat.ME",
      "subfield": "statistics"
    }
  ],
  "title": "Manifold Regression",
  "title_text": "Manifold Regression",
  "type": "preprint",
  "updated_at": "2026-10-30T12:28:03Z",
  "url": "https://research.draconitic.com/paper/1dt5jv5hqxfcq",
  "venue": "arXiv",
  "versions": []
}