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In particular, so-called slicing techniques for handling high-dimensional kernel summations profit from the simple parameter-free structure of the distance kernel. However, due to its non-smoothness in\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$x=y$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mi>x<\/mml:mi>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mi>y<\/mml:mi>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    , most of the classical theoretical results, e.g., on Wasserstein gradient flows of the corresponding MMD functional, no longer hold true. In this paper, we propose a new kernel which keeps the favorable properties of the negative distance kernel as being conditionally positive definite of order one with a nearly linear increase towards infinity and a simple slicing structure, but is Lipschitz differentiable now. Our construction is based on a simple 1D smoothing procedure of the absolute value function followed by a Riemann\u2013Liouville fractional integral transform. Numerical results demonstrate that the new kernel performs similarly well as the negative distance kernel in gradient descent methods, but now with theoretical guarantees.\n                  <\/jats:p>","DOI":"10.1007\/s10444-026-10289-5","type":"journal-article","created":{"date-parts":[[2026,3,27]],"date-time":"2026-03-27T07:17:40Z","timestamp":1774595860000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Smoothed distance kernels for MMDs and applications in Wasserstein gradient flows"],"prefix":"10.1007","volume":"52","author":[{"ORCID":"https:\/\/orcid.org\/0009-0000-1299-0580","authenticated-orcid":false,"given":"Nicolaj","family":"Rux","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6206-5705","authenticated-orcid":false,"given":"Michael","family":"Quellmalz","sequence":"additional","affiliation":[]},{"given":"Gabriele","family":"Steidl","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2026,3,27]]},"reference":[{"key":"10289_CR1","volume-title":"Handbook of mathematical functions with formulas, graphs, and mathematical tables","author":"M Abramowitz","year":"1964","unstructured":"Abramowitz, M., Stegun, I.A.: Handbook of mathematical functions with formulas, graphs, and mathematical tables, 10th edn. 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