{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,9]],"date-time":"2026-10-09T03:03:24Z","timestamp":1791515004949,"version":"4.3.4"},"reference-count":17,"publisher":"MIT Press","issue":"8","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Neural Computation"],"published-print":{"date-parts":[[2011,8]]},"abstract":"<jats:p>Mixture of experts (ME) is a modular neural network architecture for supervised classification. The double-loop expectation-maximization (EM) algorithm has been developed for learning the parameters of the ME architecture, and the iteratively reweighted least squares (IRLS) algorithm and the Newton-Raphson algorithm are two popular schemes for learning the parameters in the inner loop or gating network. In this letter, we investigate asymptotic convergence properties of the EM algorithm for ME using either the IRLS or Newton-Raphson approach. With the help of an overlap measure for the ME model, we obtain an upper bound of the asymptotic convergence rate of the EM algorithm in each case. Moreover, we find that for the Newton approach as a specific Newton-Raphson approach to learning the parameters in the inner loop, the upper bound of asymptotic convergence rate of the EM algorithm locally around the true solution \u0398* is [Formula: see text], where \u03f5&gt;0 is an arbitrarily small number, o(x) means that it is a higher-order infinitesimal as x \u2192 0, and e(\u0398*) is a measure of the average overlap of the ME model. That is, as the average overlap of the true ME model with large sample tends to zero, the EM algorithm with the Newton approach to learning the parameters in the inner loop tends to be asymptotically superlinear. Finally, we substantiate our theoretical results by simulation experiments.<\/jats:p>","DOI":"10.1162\/neco_a_00154","type":"journal-article","created":{"date-parts":[[2011,4,26]],"date-time":"2011-04-26T20:21:23Z","timestamp":1303849283000},"page":"2140-2168","source":"Crossref","is-referenced-by-count":5,"title":["Asymptotic Convergence Properties of the EM Algorithm for Mixture of Experts"],"prefix":"10.1162","volume":"23","author":[{"given":"Yan","family":"Yang","sequence":"first","affiliation":[{"name":"Department of Information Science, School of Mathematical Sciences and LMAM, Peking University, Beijing, 100871, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jinwen","family":"Ma","sequence":"additional","affiliation":[{"name":"Department of Information Science, School of Mathematical Sciences and LMAM, Peking University, Beijing, 100871, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"281","reference":[{"key":"B1","doi-asserted-by":"publisher","DOI":"10.1016\/S0893-6080(99)00043-X"},{"key":"B2","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1111\/j.2517-6161.1977.tb01600.x","volume":"39","author":"Dempster A. P.","year":"1977","journal-title":"Journal of the Royal Statistical Society, B"},{"key":"B3","volume-title":"Matrix derivatives","author":"Gerald S. R.","year":"1980"},{"key":"B4","volume-title":"Matrix analysis","author":"Horn R. A.","year":"1986"},{"key":"B5","doi-asserted-by":"publisher","DOI":"10.1162\/neco.1991.3.1.79"},{"key":"B6","first-page":"985","volume-title":"Advances in neural information processing system","volume":"4","author":"Jordan M. I.","year":"1992"},{"key":"B7","doi-asserted-by":"publisher","DOI":"10.1162\/neco.1994.6.2.181"},{"key":"B8","first-page":"1409","volume":"8","author":"Jordan M. I.","year":"1995","journal-title":"Neural Computation"},{"key":"B9","doi-asserted-by":"publisher","DOI":"10.1016\/j.patcog.2005.03.010"},{"key":"B10","doi-asserted-by":"publisher","DOI":"10.1016\/j.neucom.2004.12.009"},{"key":"B11","doi-asserted-by":"publisher","DOI":"10.1162\/089976600300014764"},{"key":"B12","doi-asserted-by":"publisher","DOI":"10.1214\/aos\/1176325371"},{"key":"B13","doi-asserted-by":"publisher","DOI":"10.1109\/TNN.2004.826217"},{"key":"B14","doi-asserted-by":"publisher","DOI":"10.1023\/A:1009627306313"},{"key":"B15","doi-asserted-by":"publisher","DOI":"10.1162\/neco.1996.8.1.129"},{"key":"B16","first-page":"633","volume-title":"Advances in neural information processing systems, 7","author":"Xu L.","year":"1994"},{"key":"B17","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-01513-7_105"}],"container-title":["Neural Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mitpressjournals.org\/doi\/pdf\/10.1162\/NECO_a_00154","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,3,5]],"date-time":"2025-03-05T08:16:05Z","timestamp":1741162565000},"score":1,"resource":{"primary":{"URL":"https:\/\/direct.mit.edu\/neco\/article\/23\/8\/2140-2168\/7691"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2011,8]]},"references-count":17,"journal-issue":{"issue":"8","published-print":{"date-parts":[[2011,8]]}},"alternative-id":["10.1162\/NECO_a_00154"],"URL":"https:\/\/doi.org\/10.1162\/neco_a_00154","relation":{},"ISSN":["0899-7667","1530-888X"],"issn-type":[{"value":"0899-7667","type":"print"},{"value":"1530-888X","type":"electronic"}],"subject":[],"published":{"date-parts":[[2011,8]]}}}