{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,1]],"date-time":"2026-04-01T11:45:25Z","timestamp":1775043925557,"version":"3.50.1"},"reference-count":27,"publisher":"MIT Press - Journals","issue":"5","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Neural Computation"],"published-print":{"date-parts":[[2016,5]]},"abstract":"<jats:p> A complex-valued convolutional network (convnet) implements the repeated application of the following composition of three operations, recursively applying the composition to an input vector of nonnegative real numbers: (1) convolution with complex-valued vectors, followed by (2) taking the absolute value of every entry of the resulting vectors, followed by (3) local averaging. For processing real-valued random vectors, complex-valued convnets can be viewed as data-driven multiscale windowed power spectra, data-driven multiscale windowed absolute spectra, data-driven multiwavelet absolute values, or (in their most general configuration) data-driven nonlinear multiwavelet packets. Indeed, complex-valued convnets can calculate multiscale windowed spectra when the convnet filters are windowed complex-valued exponentials. Standard real-valued convnets, using rectified linear units (ReLUs), sigmoidal (e.g., logistic or tanh) nonlinearities, or max pooling, for example, do not obviously exhibit the same exact correspondence with data-driven wavelets (whereas for complex-valued convnets, the correspondence is much more than just a vague analogy). Courtesy of the exact correspondence, the remarkably rich and rigorous body of mathematical analysis for wavelets applies directly to (complex-valued) convnets. <\/jats:p>","DOI":"10.1162\/neco_a_00824","type":"journal-article","created":{"date-parts":[[2016,2,19]],"date-time":"2016-02-19T00:18:45Z","timestamp":1455841125000},"page":"815-825","source":"Crossref","is-referenced-by-count":76,"title":["A Mathematical Motivation for Complex-Valued Convolutional Networks"],"prefix":"10.1162","volume":"28","author":[{"given":"Mark","family":"Tygert","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Joan","family":"Bruna","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Soumith","family":"Chintala","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yann","family":"LeCun","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Serkan","family":"Piantino","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Arthur","family":"Szlam","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"281","reference":[{"key":"B1","doi-asserted-by":"publisher","DOI":"10.1016\/j.cviu.2007.09.014"},{"key":"B2","doi-asserted-by":"publisher","DOI":"10.1109\/TPAMI.2012.230"},{"key":"B3","doi-asserted-by":"publisher","DOI":"10.1214\/14-AOS1276"},{"key":"B4","author":"Chintala S.","year":"2015","journal-title":"Scale-invariant learning and convolutional networks"},{"key":"B5","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-2544-7_9"},{"key":"B6","doi-asserted-by":"publisher","DOI":"10.1007\/978-94-011-1028-0_18"},{"key":"B7","doi-asserted-by":"publisher","DOI":"10.1109\/CVPR.2005.177"},{"key":"B8","doi-asserted-by":"publisher","DOI":"10.1137\/1.9781611970104"},{"key":"B9","doi-asserted-by":"publisher","DOI":"10.1109\/TSP.2002.808116"},{"key":"B10","first-page":"1","author":"Haensch R.","year":"2010","journal-title":"Proceedings of the 8th European Conf. EUSAR"},{"key":"B11","author":"Krizhevsky A.","year":"2009","journal-title":"Learning multiple layers of features from tiny images."},{"key":"B12","first-page":"1097","volume-title":"Advances in neural information processing systems","volume":"25","author":"Krizhevsky A.","year":"2012"},{"key":"B13","doi-asserted-by":"publisher","DOI":"10.1038\/nature14539"},{"key":"B14","doi-asserted-by":"publisher","DOI":"10.1109\/5.726791"},{"key":"B15","doi-asserted-by":"publisher","DOI":"10.1109\/ICCV.1999.790410"},{"key":"B16","doi-asserted-by":"publisher","DOI":"10.1023\/B:VISI.0000029664.99615.94"},{"key":"B17","volume-title":"A wavelet tour of signal processing: The sparse way","author":"Mallat S.","year":"2008","edition":"3"},{"key":"B18","first-page":"716","author":"Mallat S.","year":"2010","journal-title":"Proc. of the EUSIPCO Conf. 2010"},{"key":"B19","author":"Mehta P.","year":"2014","journal-title":"An exact mapping between the variational renormalization group and deep learning"},{"key":"B20","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511623820"},{"key":"B21","author":"Meyer Y.","year":"1997","journal-title":"Wavelets: Calder\u00f3n-Zygmund and multilinear operators"},{"key":"B22","doi-asserted-by":"publisher","DOI":"10.1109\/CVPR.2015.7298904"},{"key":"B23","author":"Poggio T.","year":"2012","journal-title":"The computational magic of the ventral stream: Sketch of a theory (and why some deep architectures work)"},{"key":"B24","doi-asserted-by":"crossref","DOI":"10.1561\/9781601980717","author":"Rabiner L. R.","year":"2007","journal-title":"Introduction to digital speech processing."},{"key":"B25","doi-asserted-by":"publisher","DOI":"10.1007\/BF01250288"},{"key":"B26","first-page":"444","author":"Simoncelli E. P.","year":"1995","journal-title":"Proceedings of the Internat. Conf. Image Processing 1995"},{"key":"B27","doi-asserted-by":"publisher","DOI":"10.1023\/A:1021889010444"}],"container-title":["Neural Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mitpressjournals.org\/doi\/pdf\/10.1162\/NECO_a_00824","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,3,12]],"date-time":"2021-03-12T21:41:14Z","timestamp":1615585274000},"score":1,"resource":{"primary":{"URL":"https:\/\/direct.mit.edu\/neco\/article\/28\/5\/815-825\/8157"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016,5]]},"references-count":27,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2016,5]]}},"alternative-id":["10.1162\/NECO_a_00824"],"URL":"https:\/\/doi.org\/10.1162\/neco_a_00824","relation":{},"ISSN":["0899-7667","1530-888X"],"issn-type":[{"value":"0899-7667","type":"print"},{"value":"1530-888X","type":"electronic"}],"subject":[],"published":{"date-parts":[[2016,5]]}}}