{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,19]],"date-time":"2026-02-19T12:40:29Z","timestamp":1771504829244,"version":"3.50.1"},"reference-count":38,"publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"11","license":[{"start":{"date-parts":[[2016,11,1]],"date-time":"2016-11-01T00:00:00Z","timestamp":1477958400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/ieeexplore.ieee.org\/Xplorehelp\/downloads\/license-information\/IEEE.html"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["IEEE Signal Process. Lett."],"published-print":{"date-parts":[[2016,11]]},"DOI":"10.1109\/lsp.2016.2606661","type":"journal-article","created":{"date-parts":[[2016,9,7]],"date-time":"2016-09-07T18:09:15Z","timestamp":1473271755000},"page":"1543-1546","source":"Crossref","is-referenced-by-count":32,"title":["Guaranteed Bounds on the Kullback\u2013Leibler Divergence of Univariate Mixtures"],"prefix":"10.1109","volume":"23","author":[{"given":"Frank","family":"Nielsen","sequence":"first","affiliation":[]},{"given":"Ke","family":"Sun","sequence":"additional","affiliation":[]}],"member":"263","reference":[{"key":"ref38","doi-asserted-by":"publisher","DOI":"10.1109\/LSP.2013.2288355"},{"key":"ref33","author":"bronstein","year":"2005","journal-title":"Symbolic Integration I&#x2014;Transcendental Functions (Algorithms and computation in mathematics)"},{"key":"ref32","author":"sharir","year":"1995","journal-title":"Davenport-Schinzel Sequences and Their Geometric Applications"},{"key":"ref31","doi-asserted-by":"publisher","DOI":"10.1109\/ICIP.2010.5652054"},{"key":"ref30","article-title":"PyKLGMM: Python software for computing bounds on the Kullback-Leibler divergence between mixture models","author":"nielsen","year":"2016"},{"key":"ref37","doi-asserted-by":"publisher","DOI":"10.1109\/TIT.2009.2030485"},{"key":"ref36","doi-asserted-by":"publisher","DOI":"10.1109\/TIT.2003.813506"},{"key":"ref35","doi-asserted-by":"publisher","DOI":"10.1109\/18.850703"},{"key":"ref34","doi-asserted-by":"publisher","DOI":"10.1109\/18.61115"},{"key":"ref10","doi-asserted-by":"publisher","DOI":"10.3390\/entropy-e10030200"},{"key":"ref11","doi-asserted-by":"publisher","DOI":"10.1109\/MFI.2008.4648062"},{"key":"ref12","doi-asserted-by":"publisher","DOI":"10.1109\/34.888716"},{"key":"ref13","doi-asserted-by":"publisher","DOI":"10.1587\/transinf.E92.D.2159"},{"key":"ref14","doi-asserted-by":"publisher","DOI":"10.1109\/ICASSP.2012.6289001"},{"key":"ref15","doi-asserted-by":"publisher","DOI":"10.1109\/ICASSP.2016.7472117"},{"key":"ref16","doi-asserted-by":"publisher","DOI":"10.1109\/TIT.2016.2553147"},{"key":"ref17","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-04268-3_80"},{"key":"ref18","doi-asserted-by":"publisher","DOI":"10.1109\/TIT.2011.2159046"},{"key":"ref19","first-page":"1723","article-title":"Closed-form information-theoretic divergences for statistical mixtures","author":"nielsen","year":"0","journal-title":"Proc IEEE 21st Int Conf Pattern Recognit"},{"key":"ref28","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-04245-8"},{"key":"ref4","author":"cover","year":"2012","journal-title":"Elements of Information Theory"},{"key":"ref27","doi-asserted-by":"publisher","DOI":"10.1142\/S0218195998000047"},{"key":"ref3","doi-asserted-by":"publisher","DOI":"10.1109\/NSSPW.2006.4378844"},{"key":"ref6","first-page":"746","article-title":"On Bregman Voronoi diagrams","author":"nielsen","year":"0","journal-title":"Proc 18th Annu ACM-SIAM Symp Discrete Algorithms"},{"key":"ref29","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-16007-3_1"},{"key":"ref5","first-page":"1705","article-title":"Clustering with Bregman divergences","volume":"6","author":"banerjee","year":"2005","journal-title":"J Mach Learn Res"},{"key":"ref8","doi-asserted-by":"publisher","DOI":"10.1007\/s00454-010-9256-1"},{"key":"ref7","doi-asserted-by":"publisher","DOI":"10.1109\/TIT.2009.2018176"},{"key":"ref2","doi-asserted-by":"publisher","DOI":"10.1109\/TBME.2011.2106498"},{"key":"ref9","first-page":"41","article-title":"Kullback information of normal mixture is not an analytic function","volume":"104","author":"watanabe","year":"2004","journal-title":"(in Japanese)"},{"key":"ref1","doi-asserted-by":"publisher","DOI":"10.1016\/j.amc.2008.05.130"},{"key":"ref20","doi-asserted-by":"publisher","DOI":"10.1007\/978-4-431-55978-8"},{"key":"ref22","article-title":"Statistical exponential families: A digest with flash cards","author":"nielsen","year":"2009"},{"key":"ref21","doi-asserted-by":"publisher","DOI":"10.1142\/S0219477516500139"},{"key":"ref24","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9781107279667","author":"calafiore","year":"2014","journal-title":"Optimization Model"},{"key":"ref23","article-title":"Guaranteed bounds on the Kullback-Leibler divergence of univariate mixtures using piecewise log-sum-exp inequalities","volume":"abs 1606 5850","author":"nielsen","year":"2016","journal-title":"CoRR"},{"key":"ref26","doi-asserted-by":"publisher","DOI":"10.1016\/0020-0190(95)00132-V"},{"key":"ref25","doi-asserted-by":"publisher","DOI":"10.1109\/TPAMI.2010.47"}],"container-title":["IEEE Signal Processing Letters"],"original-title":[],"link":[{"URL":"http:\/\/xplorestaging.ieee.org\/ielx7\/97\/7565698\/07562383.pdf?arnumber=7562383","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,1,12]],"date-time":"2022-01-12T16:03:18Z","timestamp":1642003398000},"score":1,"resource":{"primary":{"URL":"http:\/\/ieeexplore.ieee.org\/document\/7562383\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016,11]]},"references-count":38,"journal-issue":{"issue":"11"},"URL":"https:\/\/doi.org\/10.1109\/lsp.2016.2606661","relation":{},"ISSN":["1070-9908","1558-2361"],"issn-type":[{"value":"1070-9908","type":"print"},{"value":"1558-2361","type":"electronic"}],"subject":[],"published":{"date-parts":[[2016,11]]}}}