{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T13:42:43Z","timestamp":1787319763289,"version":"build-2736575974"},"reference-count":62,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"1","funder":[{"DOI":"10.13039\/501100001659","name":"Deutsche Forschungsgemeinschaft","doi-asserted-by":"publisher","award":["EXC-2181\/1 - 390900948"],"award-info":[{"award-number":["EXC-2181\/1 - 390900948"]}],"id":[{"id":"10.13039\/501100001659","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001659","name":"Deutsche Forschungsgemeinschaft","doi-asserted-by":"publisher","award":["SCHN 457\/17-1"],"award-info":[{"award-number":["SCHN 457\/17-1"]}],"id":[{"id":"10.13039\/501100001659","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Imaging Sci."],"published-print":{"date-parts":[[2023,3,31]]},"abstract":"<jats:p>Abstract.<\/jats:p>\n                  <jats:p>This paper introduces a novel nonlocal partial difference equation (G-PDE) for labeling metric data on graphs. The G-PDE is derived as a nonlocal reparametrization of the assignment flow approach that was introduced in [ J. Math. Imaging Vision, 58 (2017), pp. 211\u2013238]. Due to this parameterization, solving the G-PDE numerically is shown to be equivalent to computing the Riemannian gradient flow with respect to a nonconvex potential. We devise an entropy-regularized difference of convex (DC) functions decomposition of this potential and show that the basic geometric Euler scheme for integrating the assignment flow is equivalent to solving the G-PDE by an established DC programming scheme. Moreover, the viewpoint of geometric integration reveals a basic way to exploit higher-order information of the vector field that drives the assignment flow, in order to devise a novel accelerated DC programming scheme. A detailed convergence analysis of both numerical schemes is provided and illustrated by numerical experiments.<\/jats:p>","DOI":"10.1137\/22m1496141","type":"journal-article","created":{"date-parts":[[2023,3,29]],"date-time":"2023-03-29T14:01:20Z","timestamp":1680098480000},"page":"501-567","source":"Crossref","is-referenced-by-count":2,"title":["A Nonlocal Graph-PDE and Higher-Order Geometric Integration for Image Labeling"],"prefix":"10.1137","volume":"16","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0022-3891","authenticated-orcid":true,"given":"Dmitrij","family":"Sitenko","sequence":"first","affiliation":[{"name":"Image and Pattern Analysis Group, Heidelberg University, Heidelberg, 69120 Germany."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Bastian","family":"Boll","sequence":"additional","affiliation":[{"name":"Image and Pattern Analysis Group, Heidelberg University, Heidelberg, 69120 Germany."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Christoph","family":"Schn\u00f6rr","sequence":"additional","affiliation":[{"name":"Image and Pattern Analysis Group, Heidelberg University, Heidelberg, 69120 Germany."}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2023,3,30]]},"reference":[{"key":"ref1","doi-asserted-by":"publisher","DOI":"10.1137\/060669358"},{"key":"ref2","doi-asserted-by":"publisher","DOI":"10.1109\/TIP.2008.924284"},{"key":"ref3","doi-asserted-by":"publisher","DOI":"10.1137\/070698592"},{"key":"ref4","doi-asserted-by":"publisher","DOI":"10.1137\/090773908"},{"key":"ref5","doi-asserted-by":"publisher","DOI":"10.1137\/15M1022793"},{"key":"ref6","volume-title":"Deep Learning","author":"Goodfellow I.","year":"2016"},{"key":"ref7","unstructured":"R. T. Q. Chen , \nY. Rubanova , \nJ. Bettencourt , and \nD. Duvenaud , Neural ordinary differential equations, in Proceedings of Neural Information Processing Systems, 2018."},{"key":"ref8","doi-asserted-by":"publisher","DOI":"10.1561\/2200000001"},{"key":"ref9","doi-asserted-by":"publisher","DOI":"10.1007\/s11263-015-0809-x"},{"key":"ref10","doi-asserted-by":"publisher","DOI":"10.1006\/jcph.1994.1105"},{"key":"ref11","doi-asserted-by":"publisher","DOI":"10.1007\/s00032-014-0216-8"},{"key":"ref12","doi-asserted-by":"publisher","DOI":"10.1137\/16M1070426"},{"key":"ref13","doi-asserted-by":"publisher","DOI":"10.1016\/0021-9991(88)90002-2"},{"key":"ref14","doi-asserted-by":"publisher","DOI":"10.1016\/0167-2789(92)90242-F"},{"key":"ref15","doi-asserted-by":"publisher","DOI":"10.1365\/s13291-013-0066-2"},{"key":"ref16","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4939-0790-8_23"},{"key":"ref17","doi-asserted-by":"publisher","DOI":"10.1007\/s10851-016-0702-4"},{"key":"ref18","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-030-31351-7_8"},{"key":"ref19","doi-asserted-by":"publisher","DOI":"10.1007\/s41884-021-00060-8"},{"key":"ref20","doi-asserted-by":"publisher","DOI":"10.1088\/1361-6420\/ab2772"},{"key":"ref21","doi-asserted-by":"publisher","DOI":"10.1007\/s10851-019-00935-7"},{"key":"ref22","doi-asserted-by":"publisher","DOI":"10.1137\/19M1298639"},{"key":"ref23","doi-asserted-by":"publisher","DOI":"10.1007\/s10851-020-00977-2"},{"key":"ref24","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-030-75549-2_31"},{"key":"ref25","doi-asserted-by":"publisher","DOI":"10.1007\/s10851-022-01132-9"},{"key":"ref26","doi-asserted-by":"crossref","unstructured":"B. Boll , \nA. Zeilmann , \nS. Petra , and \nC. Schn\u00f6rr , Self-Certifying Classification by Linearized Deep Assignment, preprint, https:\/\/arxiv.org\/abs\/2201.11162, 2022.","DOI":"10.1002\/pamm.202200169"},{"key":"ref27","doi-asserted-by":"publisher","DOI":"10.1137\/110833294"},{"key":"ref28","doi-asserted-by":"publisher","DOI":"10.1142\/S0218202512500546"},{"key":"ref29","doi-asserted-by":"publisher","DOI":"10.1017\/S0956792520000273"},{"key":"ref30","doi-asserted-by":"publisher","DOI":"10.1007\/BF00375127"},{"key":"ref31","volume-title":"Anisotropic Diffusion in Image Processing","author":"Weickert J.","year":"1998"},{"key":"ref32","doi-asserted-by":"publisher","DOI":"10.1023\/A:1021765131316"},{"key":"ref33","doi-asserted-by":"publisher","DOI":"10.1007\/s10479-004-5022-1"},{"key":"ref34","doi-asserted-by":"publisher","DOI":"10.1137\/100818327"},{"key":"ref35","unstructured":"W. Krichene , \nA. Bayen , and \nP. Bartlett , Adaptive averaging in accelerated descent dynamics, in Proceedings of Neural Information Processing Systems, 2016."},{"key":"ref36","doi-asserted-by":"publisher","DOI":"10.1137\/17M1136845"},{"key":"ref37","doi-asserted-by":"publisher","DOI":"10.1090\/surv\/165"},{"key":"ref38","doi-asserted-by":"publisher","DOI":"10.1137\/1.9781611975628"},{"key":"ref39","doi-asserted-by":"publisher","DOI":"10.1006\/jcta.1996.0080"},{"key":"ref40","volume-title":"Spectral Graph Theory","author":"Chung F.","year":"1997"},{"key":"ref41","volume-title":"Methods of Information Geometry","author":"Amari S.-I.","year":"2000"},{"key":"ref42","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-56478-4"},{"key":"ref43","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-61860-9"},{"key":"ref44","doi-asserted-by":"publisher","DOI":"10.1007\/s11263-021-01520-5"},{"key":"ref45","doi-asserted-by":"publisher","DOI":"10.1007\/s10851-020-00986-1"},{"key":"ref46","doi-asserted-by":"publisher","DOI":"10.1080\/00207728108963798"},{"key":"ref47","doi-asserted-by":"publisher","DOI":"10.1007\/s10107-017-1180-1"},{"key":"ref48","doi-asserted-by":"publisher","DOI":"10.1137\/S1052623497318992"},{"key":"ref49","volume-title":"Numerical Optimization","author":"Nocedal J.","year":"2006","edition":"2"},{"key":"ref50","doi-asserted-by":"publisher","DOI":"10.1515\/9781400873173"},{"key":"ref51","doi-asserted-by":"publisher","DOI":"10.1016\/0041-5553(67)90040-7"},{"key":"ref52","first-page":"27","volume":"4","author":"Bauschke H. H.","year":"1997","journal-title":"J. Convex Anal."},{"key":"ref53","doi-asserted-by":"publisher","DOI":"10.1137\/17M1138558"},{"key":"ref54","doi-asserted-by":"crossref","unstructured":"D. N. Phan , \nH. M. Le , and \nH. A. L. Thi , Accelerated difference of convex functions algorithm and its application to sparse binary logistic regression, in Proceedings of the 27th International Joint Conference on Artificial Intelligence, 2018, pp. 1369\u20131375.","DOI":"10.24963\/ijcai.2018\/190"},{"key":"ref55","doi-asserted-by":"publisher","DOI":"10.1137\/1.9780898719628"},{"key":"ref56","doi-asserted-by":"publisher","DOI":"10.1007\/s10107-018-1235-y"},{"key":"ref57","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-030-92659-5_29"},{"key":"ref58","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-030-75549-2_32"},{"key":"ref59","author":"Savarino F.","year":"2021","journal-title":"CoRR"},{"key":"ref60","doi-asserted-by":"publisher","DOI":"10.2140\/pjm.1959.9.707"},{"key":"ref61","volume-title":"Kronecker Products and Matrix Calculus: With Applications","author":"Graham A.","year":"1981"},{"key":"ref62","doi-asserted-by":"publisher","DOI":"10.1137\/0803026"}],"container-title":["SIAM Journal on Imaging Sciences"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/22M1496141","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T12:52:17Z","timestamp":1787316737000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/22M1496141"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,3,30]]},"references-count":62,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2023,3,31]]}},"alternative-id":["10.1137\/22M1496141"],"URL":"https:\/\/doi.org\/10.1137\/22m1496141","relation":{},"ISSN":["1936-4954"],"issn-type":[{"value":"1936-4954","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,3,30]]}}}