{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,28]],"date-time":"2026-02-28T04:23:24Z","timestamp":1772252604402,"version":"3.50.1"},"reference-count":6,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2021,6,18]],"date-time":"2021-06-18T00:00:00Z","timestamp":1623974400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["1623190"],"award-info":[{"award-number":["1623190"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["HRD-1834620"],"award-info":[{"award-number":["HRD-1834620"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["HRD-2034030"],"award-info":[{"award-number":["HRD-2034030"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Scientific-Educational Mathematical Center of Volga Federal District","award":["075-02-2020-1478"],"award-info":[{"award-number":["075-02-2020-1478"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>One of the most effective image processing techniques is the use of convolutional neural networks that use convolutional layers. In each such layer, the value of the layer\u2019s output signal at each point is a combination of the layer\u2019s input signals corresponding to several neighboring points. To improve the accuracy, researchers have developed a version of this technique, in which only data from some of the neighboring points is processed. It turns out that the most efficient case\u2014called dilated convolution\u2014is when we select the neighboring points whose differences in both coordinates are divisible by some constant \u2113. In this paper, we explain this empirical efficiency by proving that for all reasonable optimality criteria, dilated convolution is indeed better than possible alternatives.<\/jats:p>","DOI":"10.3390\/e23060767","type":"journal-article","created":{"date-parts":[[2021,6,18]],"date-time":"2021-06-18T11:19:20Z","timestamp":1624015160000},"page":"767","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Why Dilated Convolutional Neural Networks: A Proof of Their Optimality"],"prefix":"10.3390","volume":"23","author":[{"given":"Jonatan","family":"Contreras","sequence":"first","affiliation":[{"name":"Department of Computer Science, University of Texas at El Paso, El Paso, TX 79968, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Martine","family":"Ceberio","sequence":"additional","affiliation":[{"name":"Department of Computer Science, University of Texas at El Paso, El Paso, TX 79968, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1244-1650","authenticated-orcid":false,"given":"Vladik","family":"Kreinovich","sequence":"additional","affiliation":[{"name":"Department of Computer Science, University of Texas at El Paso, El Paso, TX 79968, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,6,18]]},"reference":[{"key":"ref_1","unstructured":"Goodfellow, I., Bengio, Y., and Courville, A. (2016). Deep Leaning, MIT Press."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Li, Y., Zhang, X., and Chen, D. (2018, January 18\u201322). CSRNet: Dilated convolutional neural networks for understanding the highly congested scenes. Proceedings of the 2018 Conference on Computer Vision and Pattern Recognition CVPR\u20192018, Salt Lake City, UT, USA.","DOI":"10.1109\/CVPR.2018.00120"},{"key":"ref_3","unstructured":"Yu, F., and Koltun, V. (2016, January 2\u20134). Multi-scale context aggregation by dilated convolutions. Proceedings of the 4th International Conference on Learning Representations ICLR\u20192016, San Juan, PR, USA."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Zhang, X., Zou, Y., and Shi, W. (2017, January 23\u201325). Dilated convolution neural network with LeakyReLU for environmental sound classification. Proceedings of the 2017 22nd International Conference on Digital Signal Processing DSP\u20192017, London, UK.","DOI":"10.1109\/ICDSP.2017.8096153"},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Nguyen, H.T., and Kreinovich, V. (1997). Applications of Continuous Mathematics to Computer Science, Kluwer.","DOI":"10.1007\/978-94-017-0743-5"},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Pardalos, P., Rasskazova, V., and Vrahatis, M.N. (2021). Optimization under uncertainty explains empirical success of deep learning heuristics. Black Box Optimization, Machine Learning and No-Free Lunch Theorems, Springer.","DOI":"10.1007\/978-3-030-66515-9"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/23\/6\/767\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T06:18:11Z","timestamp":1760163491000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/23\/6\/767"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,6,18]]},"references-count":6,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2021,6]]}},"alternative-id":["e23060767"],"URL":"https:\/\/doi.org\/10.3390\/e23060767","relation":{"has-preprint":[{"id-type":"doi","id":"10.20944\/preprints202104.0501.v1","asserted-by":"object"}]},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,6,18]]}}}