{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,2]],"date-time":"2026-05-02T07:04:06Z","timestamp":1777705446492,"version":"3.51.4"},"reference-count":26,"publisher":"SAGE Publications","issue":"6","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["IFS"],"published-print":{"date-parts":[[2021,12,16]]},"abstract":"<jats:p>Manifold learning plays an important role in nonlinear dimensionality reduction. But many manifold learning algorithms cannot offer an explicit expression for dealing with the problem of out-of-sample (or new data). In recent, many improved algorithms introduce a fixed function to the object function of manifold learning for learning this expression. In manifold learning, the relationship between the high-dimensional data and its low-dimensional representation is a local homeomorphic mapping. Therefore, these improved algorithms actually change or damage the intrinsic structure of manifold learning, as well as not manifold learning. In this paper, a novel manifold learning based on polynomial approximation (PAML) is proposed, which learns the polynomial approximation of manifold learning by using the dimensionality reduction results of manifold learning and the original high-dimensional data. In particular, we establish a polynomial representation of high-dimensional data with Kronecker product, and learns an optimal transformation matrix with this polynomial representation. This matrix gives an explicit and optimal nonlinear mapping between the high-dimensional data and its low-dimensional representation, and can be directly used for solving the problem of new data. Compare with using the fixed linear or nonlinear relationship instead of the manifold relationship, our proposed method actually learns the polynomial optimal approximation of manifold learning, without changing the object function of manifold learning (i.e., keeping the intrinsic structure of manifold learning). We implement experiments over eight data sets with the advanced algorithms published in recent years to demonstrate the benefits of our algorithm.<\/jats:p>","DOI":"10.3233\/jifs-200202","type":"journal-article","created":{"date-parts":[[2021,11,2]],"date-time":"2021-11-02T16:11:02Z","timestamp":1635869462000},"page":"5791-5806","source":"Crossref","is-referenced-by-count":1,"title":["Polynomial approximation to manifold learning"],"prefix":"10.1177","volume":"41","author":[{"given":"Guo","family":"Niu","sequence":"first","affiliation":[{"name":"School of Electronics and Information Engineering, Foshan University, Foshan, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zhengming","family":"Ma","sequence":"additional","affiliation":[{"name":"School of Electronics and Information Technology, SunYat-sen University, Guangzhou, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Haoqing","family":"Chen","sequence":"additional","affiliation":[{"name":"School of Electronics and Information Technology, SunYat-sen University, Guangzhou, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xue","family":"Su","sequence":"additional","affiliation":[{"name":"School of Electronics and Information Technology, SunYat-sen University, Guangzhou, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"179","reference":[{"issue":"5","key":"10.3233\/JIFS-200202_ref1","doi-asserted-by":"crossref","first-page":"796","DOI":"10.1109\/TPAMI.2007.70735","article-title":"Riemannian manifold learning","volume":"30","author":"Lin","year":"2008","journal-title":"IEEE Transactions on Pattern Analysis and Machine Intelligence"},{"issue":"1","key":"10.3233\/JIFS-200202_ref2","doi-asserted-by":"crossref","first-page":"480","DOI":"10.1016\/j.patcog.2013.06.021","article-title":"Embedding new observation via sparse-coding for non-linear manifold learning","volume":"47","author":"Raducanu","year":"2014","journal-title":"Pattern Recognition"},{"issue":"4","key":"10.3233\/JIFS-200202_ref3","first-page":"54","article-title":"Manifold learning for visualizing and analyzing high dimensional data","volume":"25","author":"Zhang","year":"2010","journal-title":"IEEE Intelligent Systems"},{"issue":"10","key":"10.3233\/JIFS-200202_ref4","doi-asserted-by":"crossref","first-page":"4635","DOI":"10.1016\/j.csda.2008.02.031","article-title":"The out-of-sample problem for classical multi-dimensional scaling","volume":"52","author":"Trosset","year":"2008","journal-title":"Computational Statistics and Data Analysis"},{"key":"10.3233\/JIFS-200202_ref5","doi-asserted-by":"crossref","first-page":"2319","DOI":"10.1126\/science.290.5500.2319","article-title":"A global geometric framework for nonlinear dimensionality reduction","volume":"290","author":"Tenenbaum","year":"2000","journal-title":"Science"},{"issue":"3","key":"10.3233\/JIFS-200202_ref6","doi-asserted-by":"crossref","first-page":"277","DOI":"10.1111\/j.1745-3984.2003.tb01108.x","article-title":"Modern multi-dimensioanal scaling: theory and applications","volume":"40","author":"Borg","year":"2003","journal-title":"Journal of Educational Measurement"},{"key":"10.3233\/JIFS-200202_ref7","doi-asserted-by":"crossref","first-page":"2323","DOI":"10.1126\/science.290.5500.2323","article-title":"Nonlinear dimensionality reduction by locally linear embedding,","volume":"290","author":"Roweis","year":"2000","journal-title":"Science"},{"issue":"1","key":"10.3233\/JIFS-200202_ref8","doi-asserted-by":"crossref","first-page":"313","DOI":"10.1137\/S1064827502419154","article-title":"Principal manifold and nonlinear dimensionslity reduction via tangent space alignment","volume":"26","author":"Zhang","year":"2004","journal-title":"SAIM Journal on Scientific Computing"},{"issue":"6","key":"10.3233\/JIFS-200202_ref9","doi-asserted-by":"crossref","first-page":"1373","DOI":"10.1162\/089976603321780317","article-title":"Laplacian eigenmaps for dimensionality reduction and data representation","volume":"15","author":"Belkin","year":"2003","journal-title":"Neural Computation"},{"issue":"1","key":"10.3233\/JIFS-200202_ref10","doi-asserted-by":"crossref","first-page":"31","DOI":"10.1016\/j.cviu.2005.09.010","article-title":"Nonlinear manifold learning for dynamic shape and dynamic appearance","volume":"106","author":"Elganmmal","year":"2007","journal-title":"Computer Vision and Image Understanding"},{"key":"10.3233\/JIFS-200202_ref11","doi-asserted-by":"crossref","unstructured":"Yang Y. , Nie F. , et al., Local and global regressive mapping for manifold learning with out-of-sample extrapolation, AAAI Conference on Artificial Intelligence, AAAI Press, 2010, 649\u2013654.","DOI":"10.1609\/aaai.v24i1.7696"},{"issue":"1","key":"10.3233\/JIFS-200202_ref13","doi-asserted-by":"crossref","first-page":"105","DOI":"10.1007\/s10444-010-9146-3","article-title":"An alternative procedure for selecting a good value for the parameter c in RBF-interpolation","volume":"34","author":"Scheuerer","year":"2011","journal-title":"Advances in Computational Mathematics"},{"issue":"8","key":"10.3233\/JIFS-200202_ref15","doi-asserted-by":"crossref","first-page":"1236","DOI":"10.1109\/TPAMI.2006.166","article-title":"Learning nonlinear image manifold by global alignment of local linear models","volume":"28","author":"Verbeek","year":"2006","journal-title":"IEEE Transactions on Pattern Analysis and Machine Intelligence"},{"key":"10.3233\/JIFS-200202_ref16","doi-asserted-by":"crossref","unstructured":"Strange H. and Zwiggelaar R. A generalised solution to the out-of-sample extension problem in manifold learning, AAAI Conference on Artificial Intelligence, AAAI Press, 2011, 471\u2013476.","DOI":"10.1609\/aaai.v25i1.7908"},{"issue":"1","key":"10.3233\/JIFS-200202_ref17","doi-asserted-by":"crossref","first-page":"480","DOI":"10.1016\/j.patcog.2013.06.021","article-title":"Embedding new observation via sparse-coding for non-linear manifold learning","volume":"47","author":"Raducanu","year":"2014","journal-title":"Pattern Recognition"},{"issue":"9","key":"10.3233\/JIFS-200202_ref19","doi-asserted-by":"crossref","first-page":"1547","DOI":"10.1109\/TPAMI.2007.70813","article-title":"Out-of-sample extrapolation of learned manifolds","volume":"30","author":"Chin","year":"2008","journal-title":"IEEE Transactions on Pattern Analysis and Machine Intelligence"},{"issue":"3","key":"10.3233\/JIFS-200202_ref21","doi-asserted-by":"crossref","first-page":"1410","DOI":"10.1109\/TIP.2016.2520368","article-title":"Out-of-sample generalizations for supervised manifold learning for classification","volume":"25","author":"Vural","year":"2016","journal-title":"IEEE Transactions on Image Processing"},{"issue":"1","key":"10.3233\/JIFS-200202_ref22","doi-asserted-by":"crossref","first-page":"153","DOI":"10.1007\/s11390-011-9422-9","article-title":"Incremental alignment manifold learning","volume":"26","author":"Han","year":"2011","journal-title":"Journal of Computer Science and Technology"},{"issue":"1","key":"10.3233\/JIFS-200202_ref23","doi-asserted-by":"crossref","first-page":"51","DOI":"10.1109\/TSMCB.2012.2198916","article-title":"A explicit nonlinear mapping for manifold learning","volume":"43","author":"Qiao","year":"2013","journal-title":"IEEE Transactions on Cybernetics"},{"issue":"1","key":"10.3233\/JIFS-200202_ref24","doi-asserted-by":"crossref","first-page":"40","DOI":"10.1109\/TPAMI.2007.250598","article-title":"Graph embedding and extensions: a general gramework for dimensionality reduction","volume":"29","author":"Shuicheng","year":"2007","journal-title":"IEEE Transactions on Pattern Analysis and Machine Intelligence"},{"key":"10.3233\/JIFS-200202_ref27","first-page":"59","article-title":"The Laplacian eigenmaps latent variable model","volume":"2","author":"Carreira-Perpinan","year":"2007","journal-title":"Journal of Machine Learning Research"},{"issue":"3","key":"10.3233\/JIFS-200202_ref29","doi-asserted-by":"crossref","first-page":"328","DOI":"10.1109\/TPAMI.2005.55","article-title":"Face recognition using laplacianfaces","volume":"27","author":"He","year":"2005","journal-title":"IEEE Transactions on Pattern Analysis and Machine Intelligence"},{"key":"10.3233\/JIFS-200202_ref30","first-page":"1208","article-title":"Neighborhood preserving embedding,, ICCV Press,","volume":"2","author":"He","year":"2005","journal-title":"International Conference on Computer Vision"},{"key":"10.3233\/JIFS-200202_ref31","doi-asserted-by":"crossref","unstructured":"Pang Y. , Zhang L. , et al., Neighborhood preserving projections (NPP): A novel linear dimension dimension reduction method, International Conference on Computer Science, ICIC Press, 2005, 117\u2013125.","DOI":"10.1007\/11538059_13"},{"key":"10.3233\/JIFS-200202_ref39","doi-asserted-by":"crossref","unstructured":"Keyhannian S. and Nasersharif B. , Laplacian eigenmaps latent viriable model modification for Pattern Recognition International Conference on Electrical Engineering, ICEE Press, 2015, 668\u2013673.","DOI":"10.1109\/IranianCEE.2015.7146298"},{"issue":"1","key":"10.3233\/JIFS-200202_ref40","doi-asserted-by":"crossref","first-page":"77","DOI":"10.1007\/s11263-005-4939-z","article-title":"Unsupervised learning of image manifolds by semidefinite programming","volume":"70","author":"Weinberger","year":"2006","journal-title":"International Journal of Computer Vision"}],"container-title":["Journal of Intelligent &amp; Fuzzy Systems"],"original-title":[],"link":[{"URL":"https:\/\/content.iospress.com\/download?id=10.3233\/JIFS-200202","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,29]],"date-time":"2026-04-29T09:43:51Z","timestamp":1777455831000},"score":1,"resource":{"primary":{"URL":"https:\/\/journals.sagepub.com\/doi\/full\/10.3233\/JIFS-200202"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,12,16]]},"references-count":26,"journal-issue":{"issue":"6"},"URL":"https:\/\/doi.org\/10.3233\/jifs-200202","relation":{},"ISSN":["1064-1246","1875-8967"],"issn-type":[{"value":"1064-1246","type":"print"},{"value":"1875-8967","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,12,16]]}}}