2026-07-24T02:55:10Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:24276 2025-11-12T13:39:44Z ddc:004 open_access

Fast Kd-Trees for the Kullback-Leibler Divergence and Other Decomposable Bregman Divergences Pham, Tuyen Wagner, Hubert Kd-tree k-d tree nearest neighbour search Bregman divergence decomposable Bregman divergence KL divergence relative entropy cross entropy Shannon’s entropy The contributions of the paper span theoretical and implementational results. First, we prove that Kd-trees can be extended to ℝ^d with the distance measured by an arbitrary Bregman divergence. Perhaps surprisingly, this shows that the triangle inequality is not necessary for correct pruning in Kd-trees. Second, we offer an efficient algorithm and C++ implementation for nearest neighbour search for decomposable Bregman divergences. The implementation supports the Kullback-Leibler divergence (relative entropy) which is a popular distance between probability vectors and is commonly used in statistics and machine learning. This is a step toward broadening the usage of computational geometry algorithms. Our benchmarks show that our implementation efficiently handles both exact and approximate nearest neighbour queries. Compared to a linear search, we achieve two orders of magnitude speedup for practical scenarios in dimension up to 100. Our solution is simpler and more efficient than competing methods. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Tuyen Pham and Hubert Wagner 2025 Is Part Of LIPIcs, Volume 349, 19th International Symposium on Algorithms and Data Structures (WADS 2025) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.WADS.2025.45 urn:nbn:de:0030-drops-242766 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.WADS.2025.45 eng https://creativecommons.org/licenses/by/4.0/legalcode