{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,9]],"date-time":"2026-04-09T22:33:22Z","timestamp":1775774002514,"version":"3.50.1"},"reference-count":20,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2025,2,9]],"date-time":"2025-02-09T00:00:00Z","timestamp":1739059200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Shaanxi Provincial Natural Science Basic Research Program","award":["2024JC-YBMS-062"],"award-info":[{"award-number":["2024JC-YBMS-062"]}]},{"name":"Shaanxi Provincial Natural Science Basic Research Program","award":["20SKY021"],"award-info":[{"award-number":["20SKY021"]}]},{"name":"Shaanxi Provincial Natural Science Basic Research Program","award":["22SKY111"],"award-info":[{"award-number":["22SKY111"]}]},{"name":"Shaanxi Provincial Natural Science Basic Research Program","award":["23KYPY08"],"award-info":[{"award-number":["23KYPY08"]}]},{"name":"Shangluo University Foundation","award":["2024JC-YBMS-062"],"award-info":[{"award-number":["2024JC-YBMS-062"]}]},{"name":"Shangluo University Foundation","award":["20SKY021"],"award-info":[{"award-number":["20SKY021"]}]},{"name":"Shangluo University Foundation","award":["22SKY111"],"award-info":[{"award-number":["22SKY111"]}]},{"name":"Shangluo University Foundation","award":["23KYPY08"],"award-info":[{"award-number":["23KYPY08"]}]},{"name":"Shangluo University Foundation","award":["2024JC-YBMS-062"],"award-info":[{"award-number":["2024JC-YBMS-062"]}]},{"name":"Shangluo University Foundation","award":["20SKY021"],"award-info":[{"award-number":["20SKY021"]}]},{"name":"Shangluo University Foundation","award":["22SKY111"],"award-info":[{"award-number":["22SKY111"]}]},{"name":"Shangluo University Foundation","award":["23KYPY08"],"award-info":[{"award-number":["23KYPY08"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>Fuzzy matrices play a crucial role in fuzzy logic and fuzzy systems. This paper investigates the problem of supervised learning fuzzy matrices through sample pairs of input\u2013output fuzzy vectors, where the fuzzy matrix inference mechanism is based on the max\u2013min composition method. We propose an optimization approach based on stochastic gradient descent (SGD), which defines an objective function by using the mean squared error and incorporates constraints on the matrix elements (ensuring they take values within the interval [0, 1]). To address the non-smoothness of the max\u2013min composition rule, a modified smoothing function for max\u2013min is employed, ensuring stability during optimization. The experimental results demonstrate that the proposed method achieves high learning accuracy and convergence across multiple randomly generated input\u2013output vector samples.<\/jats:p>","DOI":"10.3390\/axioms14020126","type":"journal-article","created":{"date-parts":[[2025,2,10]],"date-time":"2025-02-10T03:39:47Z","timestamp":1739158787000},"page":"126","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Supervised Learning Fuzzy Matrix Based on Input\u2013Output Fuzzy Vectors"],"prefix":"10.3390","volume":"14","author":[{"given":"Meili","family":"Ye","sequence":"first","affiliation":[{"name":"School of Mathematics and Computer Application, Shangluo University, Shangluo 726000, China"}]},{"given":"Nianliang","family":"Wang","sequence":"additional","affiliation":[{"name":"School of Mathematics and Computer Application, Shangluo University, Shangluo 726000, China"}]},{"given":"Xianfeng","family":"Yu","sequence":"additional","affiliation":[{"name":"School of Mathematics and Computer Application, Shangluo University, Shangluo 726000, China"}]},{"given":"Xiao","family":"Wang","sequence":"additional","affiliation":[{"name":"School of Mathematics and Computer Application, Shangluo University, Shangluo 726000, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0220-9436","authenticated-orcid":false,"given":"Wuniu","family":"Liu","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Shaanxi Normal University, Xi\u2019an 710062, China"}]}],"member":"1968","published-online":{"date-parts":[[2025,2,9]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"338","DOI":"10.1016\/S0019-9958(65)90241-X","article-title":"Fuzzy sets","volume":"8","author":"Zadeh","year":"1965","journal-title":"Inf. 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