{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,25]],"date-time":"2026-09-25T16:26:25Z","timestamp":1790353585704,"version":"4.1.0"},"reference-count":66,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2021,7,3]],"date-time":"2021-07-03T00:00:00Z","timestamp":1625270400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100004271","name":"Sapienza Universit\u00e0 di Roma","doi-asserted-by":"publisher","award":["RG11916B88E1942F"],"award-info":[{"award-number":["RG11916B88E1942F"]}],"id":[{"id":"10.13039\/501100004271","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Variational autoencoders are deep generative models that have recently received a great deal of attention due to their ability to model the latent distribution of any kind of input such as images and audio signals, among others. A novel variational autoncoder in the quaternion domain H, namely the QVAE, has been recently proposed, leveraging the augmented second order statics of H-proper signals. In this paper, we analyze the QVAE under an information-theoretic perspective, studying the ability of the H-proper model to approximate improper distributions as well as the built-in H-proper ones and the loss of entropy due to the improperness of the input signal. We conduct experiments on a substantial set of quaternion signals, for each of which the QVAE shows the ability of modelling the input distribution, while learning the improperness and increasing the entropy of the latent space. The proposed analysis will prove that proper QVAEs can be employed with a good approximation even when the quaternion input data are improper.<\/jats:p>","DOI":"10.3390\/e23070856","type":"journal-article","created":{"date-parts":[[2021,7,4]],"date-time":"2021-07-04T22:34:36Z","timestamp":1625438076000},"page":"856","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":20,"title":["An Information-Theoretic Perspective on Proper Quaternion Variational Autoencoders"],"prefix":"10.3390","volume":"23","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-4626-4506","authenticated-orcid":false,"given":"Eleonora","family":"Grassucci","sequence":"first","affiliation":[{"name":"Department of Information Engineering, Electronics and Telecommunications, Sapienza University of Rome, Via Eudossiana 18, 00184 Rome, Italy"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4067-4504","authenticated-orcid":false,"given":"Danilo","family":"Comminiello","sequence":"additional","affiliation":[{"name":"Department of Information Engineering, Electronics and Telecommunications, Sapienza University of Rome, Via Eudossiana 18, 00184 Rome, Italy"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5793-0917","authenticated-orcid":false,"given":"Aurelio","family":"Uncini","sequence":"additional","affiliation":[{"name":"Department of Information Engineering, Electronics and Telecommunications, Sapienza University of Rome, Via Eudossiana 18, 00184 Rome, Italy"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2021,7,3]]},"reference":[{"key":"ref_1","first-page":"2672","article-title":"Generative Adversarial Nets","volume":"Volume 2","author":"Goodfellow","year":"2014","journal-title":"Advances in Neural Information Processing Systems (NIPS)"},{"key":"ref_2","unstructured":"Rezende, D.J., Metz, L., and Chintala, S. 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