{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,26]],"date-time":"2026-02-26T13:37:30Z","timestamp":1772113050258,"version":"3.50.1"},"reference-count":90,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2024,8,15]],"date-time":"2024-08-15T00:00:00Z","timestamp":1723680000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/100000006","name":"ONR","doi-asserted-by":"publisher","award":["N00014-22-1-2667"],"award-info":[{"award-number":["N00014-22-1-2667"]}],"id":[{"id":"10.13039\/100000006","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000006","name":"ONR","doi-asserted-by":"publisher","award":["DE-SC0023191"],"award-info":[{"award-number":["DE-SC0023191"]}],"id":[{"id":"10.13039\/100000006","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000006","name":"ONR","doi-asserted-by":"publisher","award":["#2019625"],"award-info":[{"award-number":["#2019625"]}],"id":[{"id":"10.13039\/100000006","id-type":"DOI","asserted-by":"publisher"}]},{"name":"DOE-MMICS SEA-CROGS","award":["N00014-22-1-2667"],"award-info":[{"award-number":["N00014-22-1-2667"]}]},{"name":"DOE-MMICS SEA-CROGS","award":["DE-SC0023191"],"award-info":[{"award-number":["DE-SC0023191"]}]},{"name":"DOE-MMICS SEA-CROGS","award":["#2019625"],"award-info":[{"award-number":["#2019625"]}]},{"name":"Zegar Family Foundation and the National Science Foundation LEAP Science and Technology Center","award":["N00014-22-1-2667"],"award-info":[{"award-number":["N00014-22-1-2667"]}]},{"name":"Zegar Family Foundation and the National Science Foundation LEAP Science and Technology Center","award":["DE-SC0023191"],"award-info":[{"award-number":["DE-SC0023191"]}]},{"name":"Zegar Family Foundation and the National Science Foundation LEAP Science and Technology Center","award":["#2019625"],"award-info":[{"award-number":["#2019625"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Maximum entropy (MaxEnt) models are a class of statistical models that use the maximum entropy principle to estimate probability distributions from data. Due to the size of modern data sets, MaxEnt models need efficient optimization algorithms to scale well for big data applications. State-of-the-art algorithms for MaxEnt models, however, were not originally designed to handle big data sets; these algorithms either rely on technical devices that may yield unreliable numerical results, scale poorly, or require smoothness assumptions that many practical MaxEnt models lack. In this paper, we present novel optimization algorithms that overcome the shortcomings of state-of-the-art algorithms for training large-scale, non-smooth MaxEnt models. Our proposed first-order algorithms leverage the Kullback\u2013Leibler divergence to train large-scale and non-smooth MaxEnt models efficiently. For MaxEnt models with discrete probability distribution of n elements built from samples, each containing m features, the stepsize parameter estimation and iterations in our algorithms scale on the order of O(mn) operations and can be trivially parallelized. Moreover, the strong \u21131 convexity of the Kullback\u2013Leibler divergence allows for larger stepsize parameters, thereby speeding up the convergence rate of our algorithms. To illustrate the efficiency of our novel algorithms, we consider the problem of estimating probabilities of fire occurrences as a function of ecological features in the Western US MTBS-Interagency wildfire data set. Our numerical results show that our algorithms outperform the state of the art by one order of magnitude and yield results that agree with physical models of wildfire occurrence and previous statistical analyses of wildfire drivers.<\/jats:p>","DOI":"10.3390\/e26080691","type":"journal-article","created":{"date-parts":[[2024,8,16]],"date-time":"2024-08-16T06:23:20Z","timestamp":1723789400000},"page":"691","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Efficient First-Order Algorithms for Large-Scale, Non-Smooth Maximum Entropy Models with Application to Wildfire Science"],"prefix":"10.3390","volume":"26","author":[{"given":"Gabriel","family":"Provencher Langlois","sequence":"first","affiliation":[{"name":"Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6672-6750","authenticated-orcid":false,"given":"Jatan","family":"Buch","sequence":"additional","affiliation":[{"name":"Department of Earth and Environmental Engineering, Columbia University, New York, NY 10027, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"J\u00e9r\u00f4me","family":"Darbon","sequence":"additional","affiliation":[{"name":"Division of Applied Mathematics, Brown University, Providence, RI 02912, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,8,15]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"620","DOI":"10.1103\/PhysRev.106.620","article-title":"Information theory and statistical mechanics","volume":"106","author":"Jaynes","year":"1957","journal-title":"Phys. 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