{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,27]],"date-time":"2026-08-27T04:19:44Z","timestamp":1787804384489,"version":"build-2784847793"},"reference-count":19,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2026,8,23]],"date-time":"2026-08-23T00:00:00Z","timestamp":1787443200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Multilayer perceptrons (MLPs) are considered as a singular model of learning machines. Singularities cause local minima and plateaus in the learning process. I\/O-equivalence, where two different MLPs are regarded as the same multivariable function, is an important concept for understanding singularities in neural networks. In this paper, I\/O-equivalence is extended to T-equivalence, which is a concept where two MLPs yield the same results through a transformation of input and output. We provide constructive families and procedures for obtaining T-equivalent networks of real-, complex-, and quaternion-valued neural networks. In particular, T-equivalence of quaternion-valued neural networks is much more complicated than that of the others.<\/jats:p>","DOI":"10.3390\/e28090946","type":"journal-article","created":{"date-parts":[[2026,8,24]],"date-time":"2026-08-24T14:49:34Z","timestamp":1787582974000},"page":"946","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Transformation Equivalence of Neural Networks"],"prefix":"10.3390","volume":"28","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4453-0615","authenticated-orcid":false,"given":"Masaki","family":"Kobayashi","sequence":"first","affiliation":[{"name":"Graduate School of Informatics, Nagoya University, Furo-cho, Chikusa-ku, Nagoya 4648601, Japan"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2026,8,23]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"589","DOI":"10.1016\/S0893-6080(05)80037-1","article-title":"Uniqueness of the weights for minimal feedforward nets with a given input-output map","volume":"5","author":"Sussmann","year":"1992","journal-title":"Neural Netw."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"317","DOI":"10.1016\/S0893-6080(00)00009-5","article-title":"Local minima and plateaus in hierarchical structures of multilayer perceptrons","volume":"13","author":"Fukumizu","year":"2000","journal-title":"Neural Netw."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1379","DOI":"10.1016\/0893-6080(95)00003-8","article-title":"Information geometry of the EM and em algorithms for neural networks","volume":"8","author":"Amari","year":"1995","journal-title":"Neural Netw."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"251","DOI":"10.1162\/089976698300017746","article-title":"Natural gradient works efficiently in learning","volume":"10","author":"Amari","year":"1998","journal-title":"Neural Comput."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"330","DOI":"10.1109\/82.142037","article-title":"Complex domain backpropagation","volume":"39","author":"Georgiou","year":"1992","journal-title":"IEEE Trans. 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