{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,31]],"date-time":"2026-03-31T02:31:01Z","timestamp":1774924261229,"version":"3.50.1"},"reference-count":47,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2025,12,27]],"date-time":"2025-12-27T00:00:00Z","timestamp":1766793600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>This study proposes a unified stochastic framework for approximating and computing the gradient of every smooth function evaluated at non-independent variables, using \u2113p-spherical distributions on Rd with d,p\u22651. The upper-bounds of the bias of the gradient surrogates do not suffer from the curse of dimensionality for any p\u22651. Additionally, the mean squared errors (MSEs) of the gradient estimators are bounded by K0N\u22121d for any p\u2208[1,2], and by K1N\u22121d2\/p when 2\u2264p\u226ad with N the sample size and K0,K1 some constants. Taking max2,log(d)&lt;p\u226ad allows for achieving dimension-free upper-bounds of MSEs. In the case where d\u226ap&lt;+\u221e, the upper-bound K2N\u22121d2\u22122\/p\/(d+2)2 is reached with K2 a constant. Such results lead to dimension-free MSEs of the proposed estimators, which boil down to estimators of the traditional gradient when the variables are independent. Numerical comparisons show the efficiency of the proposed approach.<\/jats:p>","DOI":"10.3390\/axioms15010022","type":"journal-article","created":{"date-parts":[[2025,12,29]],"date-time":"2025-12-29T00:50:42Z","timestamp":1766969442000},"page":"22","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Dimension-Free Estimators of Gradients of Functions with(out) Non-Independent Variables"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9672-9346","authenticated-orcid":false,"given":"Matieyendou","family":"Lamboni","sequence":"first","affiliation":[{"name":"Department DFR-ST, University of Guyane, 97346 Cayenne, France"},{"name":"228-UMR Espace-Dev, University of Guyane, University of R\u00e9union, IRD, University of Montpellier, 34090 Montpellier, France"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,12,27]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"470","DOI":"10.1214\/aoms\/1177729394","article-title":"Remarks on a Multivariate Transformation","volume":"23","author":"Rosenblatt","year":"1952","journal-title":"Ann. 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