{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:55:45Z","timestamp":1760151345532,"version":"build-2065373602"},"reference-count":32,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2022,3,10]],"date-time":"2022-03-10T00:00:00Z","timestamp":1646870400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/100008397","name":"The Velux Foundations","doi-asserted-by":"publisher","award":["00022924"],"award-info":[{"award-number":["00022924"]}],"id":[{"id":"10.13039\/100008397","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100009708","name":"Novo Nordisk Foundation","doi-asserted-by":"publisher","award":["NNF18OC0052000"],"award-info":[{"award-number":["NNF18OC0052000"]}],"id":[{"id":"10.13039\/501100009708","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Algorithms"],"abstract":"<jats:p>Computing sample means on Riemannian manifolds is typically computationally costly, as exemplified by computation of the Fr\u00e9chet mean, which often requires finding minimizing geodesics to each data point for each step of an iterative optimization scheme. When closed-form expressions for geodesics are not available, this leads to a nested optimization problem that is costly to solve. The implied computational cost impacts applications in both geometric statistics and in geometric deep learning. The weighted diffusion mean offers an alternative to the weighted Fr\u00e9chet mean. We show how the diffusion mean and the weighted diffusion mean can be estimated with a stochastic simulation scheme that does not require nested optimization. We achieve this by conditioning a Brownian motion in a product manifold to hit the diagonal at a predetermined time. We develop the theoretical foundation for the sampling-based mean estimation, we develop two simulation schemes, and we demonstrate the applicability of the method with examples of sampled means on two manifolds.<\/jats:p>","DOI":"10.3390\/a15030092","type":"journal-article","created":{"date-parts":[[2022,3,10]],"date-time":"2022-03-10T11:46:47Z","timestamp":1646912807000},"page":"92","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Mean Estimation on the Diagonal of Product Manifolds"],"prefix":"10.3390","volume":"15","author":[{"given":"Mathias H\u00f8jgaard","family":"Jensen","sequence":"first","affiliation":[{"name":"Department of Computer Science, University of Copenhagen, Universitetsparken 5, 2100 Copenhagen, Denmark"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6784-0328","authenticated-orcid":false,"given":"Stefan","family":"Sommer","sequence":"additional","affiliation":[{"name":"Department of Computer Science, University of Copenhagen, Universitetsparken 5, 2100 Copenhagen, Denmark"}]}],"member":"1968","published-online":{"date-parts":[[2022,3,10]]},"reference":[{"key":"ref_1","first-page":"215","article-title":"Les \u00e9l\u00e9ments al\u00e9atoires de nature quelconque dans un espace distanci\u00e9","volume":"10","year":"1948","journal-title":"Ann. 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