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In this paper, we deal with sliced optimal transport on the sphere <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\mathbb {S}^{d-1}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msup>\n                    <mml:mrow>\n                      <mml:mi>S<\/mml:mi>\n                    <\/mml:mrow>\n                    <mml:mrow>\n                      <mml:mi>d<\/mml:mi>\n                      <mml:mo>-<\/mml:mo>\n                      <mml:mn>1<\/mml:mn>\n                    <\/mml:mrow>\n                  <\/mml:msup>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> and on the rotation group <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\textrm{SO}(3)$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mtext>SO<\/mml:mtext>\n                    <mml:mo>(<\/mml:mo>\n                    <mml:mn>3<\/mml:mn>\n                    <mml:mo>)<\/mml:mo>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. We propose a parallel slicing procedure of the sphere which requires again only optimal transforms on the line. We analyze the properties of the corresponding parallelly sliced optimal transport, which provides in particular a rotationally invariant metric on the spherical probability measures. For <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\textrm{SO}(3)$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mtext>SO<\/mml:mtext>\n                    <mml:mo>(<\/mml:mo>\n                    <mml:mn>3<\/mml:mn>\n                    <mml:mo>)<\/mml:mo>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, we introduce a new two-dimensional Radon transform and develop its singular value decomposition. Based on this, we propose a sliced optimal transport on <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\textrm{SO}(3)$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mtext>SO<\/mml:mtext>\n                    <mml:mo>(<\/mml:mo>\n                    <mml:mn>3<\/mml:mn>\n                    <mml:mo>)<\/mml:mo>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. As Wasserstein distances were extensively used in barycenter computations, we derive algorithms to compute the barycenters with respect to our new sliced Wasserstein distances and provide synthetic numerical examples on the 2-sphere that demonstrate their behavior for both the free- and fixed-support setting of discrete spherical measures. In terms of computational speed, they outperform the existing methods for semicircular slicing as well as the regularized Wasserstein barycenters.<\/jats:p>","DOI":"10.1007\/s10851-024-01206-w","type":"journal-article","created":{"date-parts":[[2024,7,26]],"date-time":"2024-07-26T21:02:01Z","timestamp":1722027721000},"page":"951-976","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Parallelly Sliced Optimal Transport on Spheres and on the Rotation Group"],"prefix":"10.1007","volume":"66","author":[{"given":"Michael","family":"Quellmalz","sequence":"first","affiliation":[]},{"given":"L\u00e9o","family":"Buecher","sequence":"additional","affiliation":[]},{"given":"Gabriele","family":"Steidl","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2024,7,26]]},"reference":[{"key":"1206_CR1","volume-title":"Handbook of Mathematical Functions","year":"1972","unstructured":"Abramowitz, M., Stegun, I.A. 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