{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,23]],"date-time":"2026-02-23T03:00:01Z","timestamp":1771815601600,"version":"3.50.1"},"reference-count":25,"publisher":"Springer Science and Business Media LLC","issue":"24","license":[{"start":{"date-parts":[[2020,5,24]],"date-time":"2020-05-24T00:00:00Z","timestamp":1590278400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2020,5,24]],"date-time":"2020-05-24T00:00:00Z","timestamp":1590278400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Neural Comput &amp; Applic"],"published-print":{"date-parts":[[2020,12]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>One of the challenges in the study of generative adversarial networks (GANs) is the difficulty of its performance control. Lipschitz constraint is essential in guaranteeing training stability for GANs. Although heuristic methods such as weight clipping, gradient penalty and spectral normalization have been proposed to enforce Lipschitz constraint, it is still difficult to achieve a solution that is both practically effective and theoretically provably satisfying a Lipschitz constraint. In this paper, we introduce the boundedness and continuity (BC) conditions to enforce the Lipschitz constraint on the discriminator functions of GANs. We prove theoretically that GANs with discriminators meeting the BC conditions satisfy the Lipschitz constraint. We present a practically very effective implementation of a GAN based on a convolutional neural network (CNN) by forcing the CNN to satisfy the BC conditions (BC\u2013GAN). We show that as compared to recent techniques including gradient penalty and spectral normalization, BC\u2013GANs have not only better performances but also lower computational complexity.<\/jats:p>","DOI":"10.1007\/s00521-020-04954-z","type":"journal-article","created":{"date-parts":[[2020,5,24]],"date-time":"2020-05-24T12:02:35Z","timestamp":1590321755000},"page":"18271-18283","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":13,"title":["Lipschitz constrained GANs via boundedness and continuity"],"prefix":"10.1007","volume":"32","author":[{"given":"Kanglin","family":"Liu","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Guoping","family":"Qiu","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2020,5,24]]},"reference":[{"key":"4954_CR1","unstructured":"Anil C, Lucas J, Grosse R (2018) Sorting out lipschitz function approximation. arXiv preprint arXiv:1811.05381"},{"key":"4954_CR2","unstructured":"Arjovsky M, Chintala S, Bottou L (2017) Wasserstein gan. arXiv preprint arXiv:1701.07875"},{"key":"4954_CR3","unstructured":"Barratt S, Sharma R (2018) A note on the inception score. arXiv preprint arXiv:1801.01973"},{"key":"4954_CR4","unstructured":"Berthelot D, Schumm T, Metz L (2017) Began: boundary equilibrium generative adversarial networks. arXiv preprint arXiv:1703.10717"},{"key":"4954_CR5","unstructured":"Goodfellow I, Pouget-Abadie J, Mirza M (2014) Generative adversarial nets. 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We declare that we do not have any commercial or associative interest that represents a conflict of interest in connection with the work submitted.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}]}}