{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,24]],"date-time":"2026-10-24T00:05:50Z","timestamp":1792800350730,"version":"4.5.0"},"reference-count":35,"publisher":"Institute of Mathematical Statistics","issue":"5","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Ann. Statist."],"published-print":{"date-parts":[[2024,10,1]]},"abstract":"We provide nonasymptotic rates of convergence of the Wasserstein Generative Adversarial networks (WGAN) estimator. We build neural networks classes representing the generators and discriminators which yield a GAN that achieves the minimax optimal rate for estimating a certain probability measure \u03bc with support in Rp. The probability \u03bc is considered to be the push forward of the Lebesgue measure on the d-dimensional torus Td by a map g\u22c6:Td\u2192Rp of smoothness \u03b2+1. Measuring the error with the \u03b3-H\u00f6lder Integral Probability Metric (IPM), we obtain up to logarithmic factors, the minimax optimal rate O(n\u2212\u03b2+\u03b32\u03b2+d\u2228n\u221212) where n is the sample size, \u03b2 determines the smoothness of the target measure \u03bc, \u03b3 is the smoothness of the IPM (\u03b3=1 is the Wasserstein case) and d\u2264p is the intrinsic dimension of \u03bc. In the process, we derive a sharp interpolation inequality between H\u00f6lder IPMs. This novel result of theory of functions spaces generalizes classical interpolation inequalities to the case where the measures involved have densities on different manifolds.<\/jats:p>","DOI":"10.1214\/24-aos2430","type":"journal-article","created":{"date-parts":[[2024,11,21]],"date-time":"2024-11-21T01:27:43Z","timestamp":1732152463000},"page":"2167-2193","source":"Crossref","is-referenced-by-count":8,"title":["Wasserstein generative adversarial networks are minimax optimal distribution estimators"],"prefix":"10.1214","volume":"52","author":[{"given":"Arthur","family":"St\u00e9phanovitch","sequence":"first","affiliation":[{"name":"D\u00e9partement de Math\u00e9matiques et Applications, \u00c9cole Normale Sup\u00e9rieure, Universit\u00e9 PSL, CNRS"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Eddie","family":"Aamari","sequence":"additional","affiliation":[{"name":"D\u00e9partement de Math\u00e9matiques et Applications, \u00c9cole Normale Sup\u00e9rieure, Universit\u00e9 PSL, CNRS"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Cl\u00e9ment","family":"Levrard","sequence":"additional","affiliation":[{"name":"Institut de recherche math\u00e9matique de Rennes, Universit\u00e9 de Rennes, CNRS"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"108","reference":[{"key":"10","doi-asserted-by":"publisher","unstructured":"DIVOL, V.<\/small> (2022). 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