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The approach combines the\n                    <jats:italic>Laplace\u2013Beltrami framework<\/jats:italic>\n                    for image denoising and enhancement, introduced by Sochen, Kimmel and Malladi about 25 years ago, and the\n                    <jats:italic>assignment flow approach<\/jats:italic>\n                    introduced and studied by the authors. The sigma flow arises as the Riemannian gradient flow of generalized harmonic energies and is thus governed by a nonlinear geometric PDE which determines a harmonic map from a closed Riemannian domain manifold to a statistical manifold, equipped with the Fisher\u2013Rao metric from information geometry. A specific ingredient of the sigma flow is the mutual dependency of the Riemannian metric of the domain manifold on the evolving state. This makes the approach amenable to machine learning in a specific way, by realizing this dependency through a mapping with compact time-variant parametrization that can be learned from data. Proof-of-concept experiments demonstrate the expressivity of the sigma flow model and prediction performance. Structural similarities to transformer network architectures and networks generated by the geometric integration of sigma flows are pointed out, which highlights the connection to deep learning and, conversely, may stimulate the use of geometric design principles for structured prediction in other areas of scientific machine learning.\n                  <\/jats:p>","DOI":"10.1007\/s10851-025-01270-w","type":"journal-article","created":{"date-parts":[[2025,10,20]],"date-time":"2025-10-20T08:23:26Z","timestamp":1760948606000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Sigma Flows for Image and Data Labeling and Learning Structured Prediction"],"prefix":"10.1007","volume":"67","author":[{"ORCID":"https:\/\/orcid.org\/0009-0000-3896-9739","authenticated-orcid":false,"given":"Jonas","family":"Cassel","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3490-3350","authenticated-orcid":false,"given":"Bastian","family":"Boll","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7189-2275","authenticated-orcid":false,"given":"Stefania","family":"Petra","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7324-0609","authenticated-orcid":false,"given":"Peter","family":"Albers","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8999-2338","authenticated-orcid":false,"given":"Christoph","family":"Schn\u00f6rr","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2025,10,18]]},"reference":[{"key":"1270_CR1","doi-asserted-by":"crossref","unstructured":"Alvarez, L., Guichard, F., Lions, P.L., Morel, J.M.: Axioms and fundamental equations of image processing. 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