{"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":1774924261446,"version":"3.50.1"},"reference-count":45,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2024,6,25]],"date-time":"2024-06-25T00:00:00Z","timestamp":1719273600000},"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>Gradients of smooth functions with nonindependent variables are relevant for exploring complex models and for the optimization of the functions subjected to constraints. In this paper, we investigate new and simple approximations and computations of such gradients by making use of independent, central, and symmetric variables. Such approximations are well suited for applications in which the computations of the gradients are too expansive or impossible. The derived upper bounds of the biases of our approximations do not suffer from the curse of dimensionality for any 2-smooth function, and they theoretically improve the known results. Also, our estimators of such gradients reach the optimal (mean squared error) rates of convergence (i.e., O(N\u22121)) for the same class of functions. Numerical comparisons based on a test case and a high-dimensional PDE model show the efficiency of our approach.<\/jats:p>","DOI":"10.3390\/axioms13070426","type":"journal-article","created":{"date-parts":[[2024,6,26]],"date-time":"2024-06-26T09:29:33Z","timestamp":1719394173000},"page":"426","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Optimal and Efficient Approximations of Gradients of Functions with Nonindependent Variables"],"prefix":"10.3390","volume":"13","author":[{"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"}]}],"member":"1968","published-online":{"date-parts":[[2024,6,25]]},"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. Math. Statist."},{"key":"ref_2","first-page":"42","article-title":"D\u00e9termination des distributions dont les marges sont donn\u00e9es","volume":"225","author":"Nataf","year":"1962","journal-title":"Comptes Rendus L\u2019Acad\u00e9mie Des Sci."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Joe, H. (2014). Dependence Modeling with Copulas, Chapman & Hall\/CRC.","DOI":"10.1201\/b17116"},{"key":"ref_4","unstructured":"McNeil, A.J., Frey, R., and Embrechts, P. (2015). Quantitative Risk Management, Princeton University Press."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"51","DOI":"10.1080\/02331880500439691","article-title":"Multivariate weighted distributions: A review and some extensions","volume":"40","author":"Navarro","year":"2006","journal-title":"Statistics"},{"key":"ref_6","first-page":"229","article-title":"Fonctions de Rpartition \u00e0 n Dimensions et Leurs Marges","volume":"8","author":"Sklar","year":"1959","journal-title":"Publ. l\u2019Institut Stat. L\u2019Universit\u00e9 Paris"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"125898","DOI":"10.1016\/j.jmaa.2021.125898","article-title":"On the class of truncation invariant bivariate copulas under constraints","volume":"509","author":"Durante","year":"2022","journal-title":"J. Math. Anal. Appl."},{"key":"ref_8","first-page":"645","article-title":"On a representation of random variables","volume":"21","author":"Skorohod","year":"1976","journal-title":"Theory Probab. Appl."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"107519","DOI":"10.1016\/j.ress.2021.107519","article-title":"Multivariate sensitivity analysis and derivative-based global sensitivity measures with dependent variables","volume":"212","author":"Lamboni","year":"2021","journal-title":"Reliab. Eng. Syst. Saf."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"199","DOI":"10.1016\/j.matcom.2022.04.018","article-title":"Efficient dependency models: Simulating dependent random variables","volume":"200","author":"Lamboni","year":"2022","journal-title":"Math. Comput. Simul."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"41","DOI":"10.1007\/s11009-023-09993-2","article-title":"On exact distribution for multivariate weighted distributions and classification","volume":"25","author":"Lamboni","year":"2023","journal-title":"Methodol. Comput. Appl. Probab."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Lamboni, M. (2024). Measuring inputs-outputs association for time-dependent hazard models under safety objectives using kernels. Int. J. Uncertain. Quantif., 1\u201317.","DOI":"10.1615\/Int.J.UncertaintyQuantification.2024049119"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"218","DOI":"10.1016\/j.ress.2017.06.001","article-title":"Sobol\u2019 indices for problems defined in non-rectangular domains","volume":"167","author":"Kucherenko","year":"2017","journal-title":"Reliab. Eng. Syst. Saf."},{"key":"ref_14","unstructured":"Lamboni, M. (2021). On dependency models and dependent generalized sensitivity indices. arXiv."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Lamboni, M. (2023). Derivative Formulas and Gradient of Functions with Non-Independent Variables. Axioms, 12.","DOI":"10.3390\/axioms12090845"},{"key":"ref_16","unstructured":"Nemirovsky, A., and Yudin, D. (1983). Problem Complexity and Method Efficiency in Optimization, Wiley & Sons."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"172","DOI":"10.1002\/nme.2687","article-title":"Monte Carlo gradient estimation in high dimensions","volume":"81","author":"Patelli","year":"2010","journal-title":"Int. J. Numer. Methods Eng."},{"key":"ref_18","unstructured":"Agarwal, A., Dekel, O., and Xiao, L. (2010, January 27\u201329). Optimal Algorithms for Online Convex Optimization with Multi-Point Bandit Feedback. Proceedings of the The 23rd Conference on Learning Theory, COLT 2010, Haifa, Israel."},{"key":"ref_19","unstructured":"Bach, F., and Perchet, V. (2016, January 23\u201326). Highly-Smooth Zero-th Order Online Optimization. Proceedings of the 29th Annual Conference on Learning Theory, New York, NY, USA."},{"key":"ref_20","unstructured":"Akhavan, A., Pontil, M., and Tsybakov, A.B. (2024, June 18). Exploiting Higher Order Smoothness in Derivative-Free Optimization and Continuous Bandits. Available online: https:\/\/arxiv.org\/abs\/2006.07862."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"3009","DOI":"10.1016\/j.matcom.2009.01.023","article-title":"Derivative based global sensitivity measures and the link with global sensitivity indices","volume":"79","author":"Sobol","year":"2009","journal-title":"Math. Comput. Simul."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"1135","DOI":"10.1016\/j.ress.2008.05.006","article-title":"Monte Carlo evaluation of derivative-based global sensitivity measures","volume":"94","author":"Kucherenko","year":"2009","journal-title":"Reliab. Eng. Syst. Saf."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"45","DOI":"10.1016\/j.matcom.2013.02.002","article-title":"Derivative-based global sensitivity measures: General links with Sobol\u2019 indices and numerical tests","volume":"87","author":"Lamboni","year":"2013","journal-title":"Math. Comput. Simul."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"105","DOI":"10.1016\/j.matcom.2014.05.005","article-title":"Crossed-derivative based sensitivity measures for interaction screening","volume":"105","author":"Roustant","year":"2014","journal-title":"Math. Comput. Simul."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"212","DOI":"10.1016\/j.jspi.2013.11.007","article-title":"Total interaction index: A variance-based sensitivity index for second-order interaction screening","volume":"147","author":"Fruth","year":"2014","journal-title":"J. Stat. Plan. Inference"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"236","DOI":"10.1016\/j.matcom.2019.10.017","article-title":"Derivative-based generalized sensitivity indices and Sobol\u2019 indices","volume":"170","author":"Lamboni","year":"2020","journal-title":"Math. Comput. Simul."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"137","DOI":"10.1016\/j.matcom.2020.08.006","article-title":"Derivative-based integral equalities and inequality: A proxy-measure for sensitivity analysis","volume":"179","author":"Lamboni","year":"2021","journal-title":"Math. Comput. Simul."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"691","DOI":"10.1016\/j.matcom.2021.12.019","article-title":"Weak derivative-based expansion of functions: ANOVA and some inequalities","volume":"194","author":"Lamboni","year":"2022","journal-title":"Math. Comput. Simul."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"1903","DOI":"10.1214\/aop\/1022677553","article-title":"Isoperimetric and Analytic Inequalities for Log-Concave Probability Measures","volume":"27","author":"Bobkov","year":"1999","journal-title":"Ann. Probab."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"3081","DOI":"10.1214\/17-EJS1310","article-title":"Poincar\u00e9 inequalities on intervals-application to sensitivity analysis","volume":"11","author":"Roustant","year":"2017","journal-title":"Electron. J. Statist."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"97","DOI":"10.3402\/tellusa.v38i2.11706","article-title":"Variational algorithms for analysis and assimilation of meteorological observations: Theoretical aspects","volume":"38","author":"Talagrand","year":"1986","journal-title":"Tellus A Dyn. Meteorol. Oceanogr."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"245","DOI":"10.2151\/jmsj1965.75.1B_245","article-title":"Sensitivity analysis in variational data assimilation","volume":"75","author":"Ngodock","year":"1997","journal-title":"J.-Meteorol. Soc. Jpn."},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Cacuci, D.G. (2005). Sensitivity and Uncertainty Analysis\u2014Theory, Chapman & Hall\/CRC.","DOI":"10.1201\/9780203483572"},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Gunzburger, M.D. (2003). Perspectives in Flow Control and Optimization, SIAM.","DOI":"10.1137\/1.9780898718720"},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Borzi, A., and Schulz, V. (2012). Computational Optimization of Systems Governed by Partial Differential Equations, SIAM.","DOI":"10.1137\/1.9781611972054"},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Ghanem, R., Higdon, D., and Owhadi, H. (2017). Handbook of Uncertainty Quantification, Springer International Publishing.","DOI":"10.1007\/978-3-319-12385-1"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"1","DOI":"10.18637\/jss.v104.i05","article-title":"Calculus: High-Dimensional Numerical and Symbolic Calculus in R","volume":"104","author":"Guidotti","year":"2022","journal-title":"J. Stat. Softw."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"4117","DOI":"10.1175\/2007MWR1904.1","article-title":"Comparing Adjoint- and Ensemble-Sensitivity Analysis with Applications to Observation Targeting","volume":"135","author":"Ancell","year":"2007","journal-title":"Mon. Weather Rev."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"627","DOI":"10.1007\/s00466-006-0127-9","article-title":"Relative importance of uncertain structural parameters. Part I: Algorithm","volume":"40","author":"Pradlwarter","year":"2007","journal-title":"Comput. Mech."},{"key":"ref_40","first-page":"45","article-title":"Optimal accuracy orders of stochastic approximation algorithms","volume":"26","author":"Polyak","year":"1990","journal-title":"Probl. Peredachi Inf."},{"key":"ref_41","unstructured":"Zemanian, A. (1987). Distribution Theory and Transform Analysis: An Introduction to Generalized Functions, with Applications, Dover Publications. Dover Books on Advanced Mathematics."},{"key":"ref_42","unstructured":"Strichartz, R. (1994). A Guide to Distribution Theory and Fourier Transforms, Studies in Advanced Mathematics; CRC Press."},{"key":"ref_43","doi-asserted-by":"crossref","unstructured":"Lamboni, M. (2024). Kernel-based Measures of Association Between Inputs and Outputs Using ANOVA. Sankhya A.","DOI":"10.1007\/s13171-024-00354-w"},{"key":"ref_44","unstructured":"Gilbert, P., and Varadhan, R. (2024, June 18). R-Package numDeriv: Accurate Numerical Derivatives; CRAN Repository. Available online: http:\/\/optimizer.r-forge.r-project.org\/."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"1","DOI":"10.18637\/jss.v033.i09","article-title":"Solving Differential Equations in R: Package deSolve","volume":"33","author":"Soetaert","year":"2010","journal-title":"J. Stat. Softw."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/7\/426\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T15:04:09Z","timestamp":1760108649000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/7\/426"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,6,25]]},"references-count":45,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2024,7]]}},"alternative-id":["axioms13070426"],"URL":"https:\/\/doi.org\/10.3390\/axioms13070426","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,6,25]]}}}