{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,31]],"date-time":"2026-03-31T13:37:51Z","timestamp":1774964271673,"version":"3.50.1"},"reference-count":13,"publisher":"MIT Press - Journals","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Neural Computation"],"published-print":{"date-parts":[[2005,1,1]]},"abstract":"<jats:p> In this letter, we provide a study of learning in a Hilbert space of vector-valued functions. We motivate the need for extending learning theory of scalar-valued functions by practical considerations and establish some basic results for learning vector-valued functions that should prove useful in applications. Specifically, we allow an output space Y to be a Hilbert space, and we consider a reproducing kernel Hilbert space of functions whose values lie in Y. In this setting, we derive the form of the minimal norm interpolant to a finite set of data and apply it to study some regularization functionals that are important in learning theory. We consider specific examples of such functionals corresponding to multiple-output regularization networks and support vector machines, for both regression and classification. Finally, we provide classes of operator-valued kernels of the dot product and translation-invariant type. <\/jats:p>","DOI":"10.1162\/0899766052530802","type":"journal-article","created":{"date-parts":[[2004,11,25]],"date-time":"2004-11-25T03:13:05Z","timestamp":1101352385000},"page":"177-204","source":"Crossref","is-referenced-by-count":304,"title":["On Learning Vector-Valued Functions"],"prefix":"10.1162","volume":"17","author":[{"given":"Charles A.","family":"Micchelli","sequence":"first","affiliation":[{"name":"Department of Mathematics and Statistics, State University of New York, University at Albany, Albany, NY, 12222, U.S.A."}]},{"given":"Massimiliano","family":"Pontil","sequence":"additional","affiliation":[{"name":"Department of Computer Sciences, University College London, London WC1E, England, UK,"}]}],"member":"281","reference":[{"key":"p_3","doi-asserted-by":"publisher","DOI":"10.2307\/1990404"},{"key":"p_6","first-page":"722","volume":"3","author":"Bennett K.P.","year":"1993","journal-title":"Optimization Methods and Software"},{"key":"p_8","doi-asserted-by":"publisher","DOI":"10.1126\/science.272.5270.1905"},{"key":"p_9","doi-asserted-by":"publisher","DOI":"10.1111\/1467-9868.00054"},{"key":"p_12","doi-asserted-by":"publisher","DOI":"10.1111\/j.1467-9868.2004.02054.x"},{"key":"p_13","doi-asserted-by":"publisher","DOI":"10.1162\/15324430260185628"},{"key":"p_17","doi-asserted-by":"publisher","DOI":"10.1023\/A:1018946025316"},{"key":"p_19","doi-asserted-by":"publisher","DOI":"10.1016\/0024-3795(93)00232-O"},{"key":"p_20","doi-asserted-by":"publisher","DOI":"10.1109\/5254.736001"},{"key":"p_23","doi-asserted-by":"publisher","DOI":"10.1137\/0716007"},{"key":"p_27","doi-asserted-by":"publisher","DOI":"10.1162\/15324430260185556"},{"key":"p_29","first-page":"145","volume":"1","author":"Sejnowski T.J.","year":"1987","journal-title":"Complex Systems"},{"key":"p_35","doi-asserted-by":"publisher","DOI":"10.1109\/34.735807"}],"container-title":["Neural Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mitpressjournals.org\/doi\/pdf\/10.1162\/0899766052530802","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,3,12]],"date-time":"2021-03-12T21:51:12Z","timestamp":1615585872000},"score":1,"resource":{"primary":{"URL":"https:\/\/direct.mit.edu\/neco\/article\/17\/1\/177-204\/6909"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2005,1,1]]},"references-count":13,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2005,1,1]]}},"alternative-id":["10.1162\/0899766052530802"],"URL":"https:\/\/doi.org\/10.1162\/0899766052530802","relation":{},"ISSN":["0899-7667","1530-888X"],"issn-type":[{"value":"0899-7667","type":"print"},{"value":"1530-888X","type":"electronic"}],"subject":[],"published":{"date-parts":[[2005,1,1]]}}}