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However, the inference and learning problems in the Boltzmann machine are NP-hard. The investigation of an effective learning algorithm for the Boltzmann machine is one of the most important challenges in the field of statistical machine learning. In this paper, we study Boltzmann machine learning based on the (first-order) spatial Monte Carlo integration method, referred to as the 1-SMCI learning method, which was proposed in the author’s previous paper. In the first part of this paper, we compare the method with the maximum pseudo-likelihood estimation (MPLE) method using a theoretical and a numerical approaches, and show the 1-SMCI learning method is more effective than the MPLE. In the latter part, we compare the 1-SMCI learning method with other effective methods, ratio matching and minimum probability flow, using a numerical experiment, and show the 1-SMCI learning method outperforms them.</jats:p>"}]}],"creator":[{"@id":"https://cir.nii.ac.jp/crid/1380004239471228672","@type":"Researcher","foaf:name":[{"@value":"Muneki Yasuda"}],"jpcoar:affiliationName":[{"@value":"Graduate School of Science and Engineering, Yamagata University, Yamagata 992-8510, Japan"}]}],"publication":{"publicationIdentifier":[{"@type":"EISSN","@value":"19994893"}],"prism:publicationName":[{"@value":"Algorithms"}],"dc:publisher":[{"@value":"MDPI AG"}],"prism:publicationDate":"2018-04-04","prism:volume":"11","prism:number":"4","prism:startingPage":"42"},"reviewed":"false","dcterms:accessRights":"http://purl.org/coar/access_right/c_abf2","dc:rights":["https://creativecommons.org/licenses/by/4.0/"],"url":[{"@id":"https://www.mdpi.com/1999-4893/11/4/42/pdf"}],"createdAt":"2018-04-04","modifiedAt":"2025-10-11","foaf:topic":[{"@id":"https://cir.nii.ac.jp/all?q=machine%20learning;%20Boltzmann%20machine;%20Monte%20Carlo%20integration;%20approximate%20algorithm","dc:title":"machine learning; Boltzmann machine; Monte Carlo integration; approximate algorithm"},{"@id":"https://cir.nii.ac.jp/all?q=Boltzmann%20machine","dc:title":"Boltzmann machine"},{"@id":"https://cir.nii.ac.jp/all?q=Industrial%20engineering.%20Management%20engineering","dc:title":"Industrial engineering. 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