{
  "abstract": "Rectified flow (RF) is an ODE-based generative modeling framework that has achieved remarkable success in practice over the past few years. By design, RF seeks to produce flows that have a high degree of straightness, in the sense that the resulting ODE trajectories can be obtained with relatively few (ideally, one) sampling steps. However, a rigorous analysis of straightness or near-straightness of RF flows remains missing in the literature. In this paper, we study settings in which the coupling generated by two-step RF is provably straight at the population level. We provide a complete characterization in two popular classes of data distributions: (i) smooth, strongly log-concave densities and (ii) Gaussian mixture models. Our main results yield sharp conditions for the coupling generated by two-step RF to be straight. Non-straight couplings are constructed when these conditions fail, which demonstrate the sharpness of these conditions. Using the theory developed on the straightness of two-step RF, we propose a notion of approximate straightness that measures the straightness of the learned couplings and can be estimated from the data. We describe a procedure for constructing asymptotic confidence intervals for our proposed notion of approximate straightness and propose a data-driven method for selecting the number of reflow rounds in multiple iterations of RF.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.36090",
  "author_total": 4,
  "authors": [
    {
      "name": "Tuan Pham"
    },
    {
      "name": "Saptarshi Roy"
    },
    {
      "name": "Alessandro Rinaldo"
    },
    {
      "name": "Purnamrita Sarkar"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5w872sz2t1a4w.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.36090",
  "fulltext_url": null,
  "id": "5w872sz2t1a4w",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.36090",
  "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",
      "math.ST"
    ],
    [
      "arxiv",
      "stat.TH"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.36090",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5w872sz2t1a4w.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:53Z",
      "native_id": "2610.36090",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.36090"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.ST",
      "subfield": "statistics"
    }
  ],
  "title": "Setting Things Straight: On the Straightness and Non-Straightness of Two-Step Rectified Flow",
  "title_text": "Setting Things Straight: On the Straightness and Non-Straightness of Two-Step Rectified Flow",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:35Z",
  "url": "https://research.draconitic.com/paper/5w872sz2t1a4w",
  "venue": "arXiv",
  "versions": []
}