{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,30]],"date-time":"2025-10-30T07:17:08Z","timestamp":1761808628887,"version":"build-2065373602"},"reference-count":29,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2023,10,10]],"date-time":"2023-10-10T00:00:00Z","timestamp":1696896000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"DFG, German Research Foundation","award":["416228727\u2014SFB 1410"],"award-info":[{"award-number":["416228727\u2014SFB 1410"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>The quantization problem aims to find the best possible approximation of probability measures on\u00a0Rd\u00a0using finite and discrete measures. The Wasserstein distance is a typical choice to measure the quality of the approximation. This contribution investigates the properties and robustness of the entropy-regularized quantization problem, which relaxes the standard quantization problem. The proposed approximation technique naturally adopts the softmin function, which is well known for its robustness from both theoretical and practicability standpoints. Moreover, we use the entropy-regularized Wasserstein distance to evaluate the quality of the soft quantization problem\u2019s approximation, and we implement a stochastic gradient approach to achieve the optimal solutions. The control parameter in our proposed method allows for the adjustment of the optimization problem\u2019s difficulty level, providing significant advantages when dealing with exceptionally challenging problems of interest. As well, this contribution empirically illustrates the performance of the method in various expositions.<\/jats:p>","DOI":"10.3390\/e25101435","type":"journal-article","created":{"date-parts":[[2023,10,11]],"date-time":"2023-10-11T02:11:01Z","timestamp":1696990261000},"page":"1435","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Soft Quantization Using Entropic Regularization"],"prefix":"10.3390","volume":"25","author":[{"ORCID":"https:\/\/orcid.org\/0009-0006-3273-9063","authenticated-orcid":false,"given":"Rajmadan","family":"Lakshmanan","sequence":"first","affiliation":[{"name":"Faculty of Mathematics, Technische Universit\u00e4t Chemnitz, D-09111 Chemnitz, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8876-2429","authenticated-orcid":false,"given":"Alois","family":"Pichler","sequence":"additional","affiliation":[{"name":"Faculty of Mathematics, Technische Universit\u00e4t Chemnitz, D-09111 Chemnitz, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,10,10]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"147","DOI":"10.1090\/conm\/094\/1012985","article-title":"A Classification of Disintegrations of Measures","volume":"94","author":"Graf","year":"1989","journal-title":"Contemp. 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