{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,26]],"date-time":"2026-08-26T15:29:28Z","timestamp":1787758168128,"version":"build-2784847793"},"reference-count":24,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2024,3,29]],"date-time":"2024-03-29T00:00:00Z","timestamp":1711670400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"European Regional Development Fund","award":["EXCELLENCE\/1216\/0296"],"award-info":[{"award-number":["EXCELLENCE\/1216\/0296"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>The main focus of this paper is the derivation of the structural properties of the test channels of Wyner\u2019s operational information rate distortion function (RDF), R\u00af(\u0394X), for arbitrary abstract sources and, subsequently, the derivation of additional properties for a tuple of multivariate correlated, jointly independent, and identically distributed Gaussian random variables, {Xt,Yt}t=1\u221e, Xt:\u03a9\u2192Rnx, Yt:\u03a9\u2192Rny, with average mean-square error at the decoder and the side information, {Yt}t=1\u221e, available only at the decoder. For the tuple of multivariate correlated Gaussian sources, we construct optimal test channel realizations which achieve the informational RDF, R\u00af(\u0394X)=\u25b5infM(\u0394X)I(X;Z|Y), where M(\u0394X) is the set of auxiliary RVs Z such that PZ|X,Y=PZ|X, X^=f(Y,Z), and E{||X\u2212X^||2}\u2264\u0394X. We show the following fundamental structural properties: (1) Optimal test channel realizations that achieve the RDF and satisfy conditional independence, PX|X^,Y,Z=PX|X^,Y=PX|X^,EX|X^,Y,Z=EX|X^=X^. (2) Similarly, for the conditional RDF, RX|Y(\u0394X), when the side information is available to both the encoder and the decoder, we show the equality R\u00af(\u0394X)=RX|Y(\u0394X). (3) We derive the water-filling solution for RX|Y(\u0394X).<\/jats:p>","DOI":"10.3390\/e26040306","type":"journal-article","created":{"date-parts":[[2024,3,31]],"date-time":"2024-03-31T13:39:37Z","timestamp":1711892377000},"page":"306","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Structural Properties of the Wyner\u2013Ziv Rate Distortion Function: Applications for Multivariate Gaussian Sources"],"prefix":"10.3390","volume":"26","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2505-4569","authenticated-orcid":false,"given":"Michail","family":"Gkagkos","sequence":"first","affiliation":[{"name":"Department of Electrical and Computer Engineering, Texas A & M University, College Station, TX 77843, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2168-0231","authenticated-orcid":false,"given":"Charalambos D.","family":"Charalambous","sequence":"additional","affiliation":[{"name":"Department of Electrical and Computer Engineering, University of Cyprus, P.O. Box 20537, CY-1678 Nicosia, Cyprus"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2024,3,29]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1109\/TIT.1976.1055508","article-title":"The rate-distortion function for source coding with side information at the decoder","volume":"22","author":"Wyner","year":"1976","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"60","DOI":"10.1016\/S0019-9958(78)90034-7","article-title":"The rate-distortion function for source coding with side information at the decoder-II: General sources","volume":"38","author":"Wyner","year":"1978","journal-title":"Inf. Control."},{"key":"ref_3","unstructured":"Berger, T. (1971). 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