{"id":3030,"date":"2025-04-11T07:02:33","date_gmt":"2025-04-11T07:02:33","guid":{"rendered":"https:\/\/mailitics.com\/index.php\/2025\/04\/11\/2504-07820\/"},"modified":"2025-04-11T07:02:33","modified_gmt":"2025-04-11T07:02:33","slug":"2504-07820","status":"publish","type":"post","link":"https:\/\/mailitics.com\/index.php\/2025\/04\/11\/2504-07820\/","title":{"rendered":"Smoothed Distance Kernels for MMDs and Applications in Wasserstein Gradient Flows"},"content":{"rendered":"

Smoothed Distance Kernels for MMDs and Applications in Wasserstein Gradient Flows
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arXiv:2504.07820v1 Announce Type: new
\nAbstract: Negative distance kernels $K(x,y) := – |x-y|$ were used in the definition of maximum mean discrepancies (MMDs) in statistics and lead to favorable numerical results in various applications. 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 $x=y$, most of the classical theoretical results, e.g. on Wasserstein gradient flows of the corresponding MMD functional do not 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-Liouville 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.<\/div>\n

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\n Nicolaj Rux, Michael Quellmalz, Gabriele Steidl
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Smoothed Distance Kernels for MMDs and Applications in Wasserstein Gradient Flows arXiv:2504.07820v1 Announce Type: new Abstract: Negative distance kernels $K(x,y) := – |x-y|$ were used in the definition of maximum mean discrepancies (MMDs) in statistics and lead to favorable numerical results in various applications. In particular, so-called slicing techniques for handling high-dimensional kernel summations profit […]<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[62,113,1172,420,112],"tags":[2359,379,1135],"class_list":["post-3030","post","type-post","status-publish","format-standard","hentry","category-aimldsaimlds","category-cs-lg","category-math-fa","category-math-pr","category-stat-ml","tag-distance","tag-gradient","tag-kernel"],"_links":{"self":[{"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/posts\/3030"}],"collection":[{"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/comments?post=3030"}],"version-history":[{"count":0,"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/posts\/3030\/revisions"}],"wp:attachment":[{"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/media?parent=3030"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/categories?post=3030"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/tags?post=3030"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}