{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,10]],"date-time":"2026-06-10T16:40:48Z","timestamp":1781109648458,"version":"3.54.1"},"reference-count":26,"publisher":"IGI Global Scientific Publishing","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2015,4,1]]},"abstract":"<p>The Local Riemannian Manifold Learning (LRML) recovers the manifold topology and geometry behind database samples through normal coordinate neighborhoods computed by the exponential map. Besides, LRML uses barycentric coordinates to go from the parameter space to the Riemannian manifold in order to perform the manifold synthesis. Despite of the advantages of LRML, the obtained parameterization cannot be used as a representational space without ambiguities. Besides, the synthesis process needs a simplicial decomposition of the lower dimensional domain to be efficiently performed, which is not considered in the LRML proposal. In this paper, the authors address these drawbacks of LRML by using a composition procedure to combine the normal coordinate neighborhoods for building a suitable representational space. Moreover, they incorporate a polyhedral geometry framework to the LRML method to give an efficient background for the synthesis process and data analysis. In the computational experiments, the authors verify the efficiency of the LRML combined with the composition and discrete geometry frameworks for dimensionality reduction, synthesis and data exploration.<\/p>","DOI":"10.4018\/ijncr.2015040103","type":"journal-article","created":{"date-parts":[[2015,4,9]],"date-time":"2015-04-09T09:31:51Z","timestamp":1428571911000},"page":"37-68","source":"Crossref","is-referenced-by-count":1,"title":["Composition of Local Normal Coordinates and Polyhedral Geometry in Riemannian Manifold Learning"],"prefix":"10.4018","volume":"5","author":[{"given":"Gast\u00e3o F.","family":"Miranda Jr.","sequence":"first","affiliation":[{"name":"Department of Mathematics, Federal University of Sergipe, Sao Cristovao, Brazil"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Gilson","family":"Giraldi","sequence":"additional","affiliation":[{"name":"National Laboratory for Scientific Computing, Petropolis, Brazil"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Carlos E.","family":"Thomaz","sequence":"additional","affiliation":[{"name":"Department of Electrical Engineering, University Center of FEI, Sao Bernardo do Campo, Brazil"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Daniel","family":"Mill\u00e0n","sequence":"additional","affiliation":[{"name":"Polytechnic University of Catalonia-Barcelona, Barcelona, Spain"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"2432","reference":[{"key":"ijncr.2015040103-0","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-642-61257-2","author":"E. L.Allgower","year":"1990","journal-title":"Numerical Continuation Methods: An Introduction"},{"issue":"10","key":"ijncr.2015040103-1","doi-asserted-by":"crossref","first-page":"2385","DOI":"10.1162\/089976600300014980","article-title":"Generalized discriminant analysis using a kernel approach.","volume":"12","author":"G.Baudat","year":"2000","journal-title":"Neural Computation"},{"key":"ijncr.2015040103-2","doi-asserted-by":"crossref","first-page":"1373","DOI":"10.1162\/089976603321780317","article-title":"Laplacian eigenmaps for dimensionality reduction and data representation.","volume":"15","author":"M.Belkin","year":"2003","journal-title":"Neural Computation"},{"key":"ijncr.2015040103-3","doi-asserted-by":"crossref","first-page":"208","DOI":"10.1145\/1542362.1542403","article-title":"Incremental construction of the delaunay triangulation and the delaunay graph in medium dimension.","author":"J.-D.Boissonnat","year":"2009","journal-title":"Proceedings of the Twenty-fifth Annual Symposium on Computational Geometry"},{"key":"ijncr.2015040103-4","doi-asserted-by":"crossref","unstructured":"Brun, A., Westin, C., Herberthson, M., & Knutsson, H. (2005). Fast manifold learning based on riemannian normal coordinates. In Image Analysis, volume 3540 of LNCS, pages 920\u2013929. Springer.","DOI":"10.1007\/11499145_93"},{"key":"ijncr.2015040103-5","author":"T. F.Cox","year":"2001","journal-title":"Multidimensional scaling"},{"key":"ijncr.2015040103-6","author":"M.do Carmo","year":"1988","journal-title":"Geometria Riemanniana"},{"key":"ijncr.2015040103-7","author":"B.Dubrovin","year":"1992","journal-title":"Modern geometry: Methods and Applications"},{"key":"ijncr.2015040103-8","unstructured":"Engel, D., H\u00fcttenberger, L., & Hamann, B. (2012). A Survey of Dimension Reduction Methods for High-dimensional Data Analysis and Visualization. In Proceedings of IRTG 1131 Workshop 2011, volume 27, pages 135\u2013149, Germany. Schloss Dagstuhl."},{"key":"ijncr.2015040103-9","first-page":"1909","article-title":"Manifold learning: The price of normalization.","volume":"9","author":"Y.Goldberg","year":"2008","journal-title":"Journal of Machine Learning Research"},{"key":"ijncr.2015040103-10","doi-asserted-by":"crossref","DOI":"10.1007\/978-0-387-21606-5","author":"T.Hastie","year":"2001","journal-title":"The Elements of Statistical Learning"},{"key":"ijncr.2015040103-11","article-title":"Aprendizagem e s\u00edntese de variedades via coordenadas normais de riemann locais e baricentricas.","author":"G. F. M.Junior","year":"2013","journal-title":"Proc. of the ENIAC"},{"key":"ijncr.2015040103-12","doi-asserted-by":"crossref","DOI":"10.1007\/978-0-387-39351-3","author":"J. A.Lee","year":"2007","journal-title":"Nonlinear Dimensionality Reduction"},{"issue":"5","key":"ijncr.2015040103-13","doi-asserted-by":"crossref","first-page":"796","DOI":"10.1109\/TPAMI.2007.70735","article-title":"Riemannian Manifold Learning.","volume":"30","author":"T.Lin","year":"2008","journal-title":"IEEE Transactions on Pattern Analysis and Machine Intelligence"},{"issue":"5","key":"ijncr.2015040103-14","article-title":"Riemannian manifold learning.","volume":"30","author":"T.Lin","year":"2008","journal-title":"IEEE Transactions on Pattern Analysis and Machine Intelligence"},{"issue":"1","key":"ijncr.2015040103-15","doi-asserted-by":"crossref","first-page":"55","DOI":"10.1109\/MSP.2013.2279894","article-title":"Manifold-learning-based feature extraction for classification of hyperspectral data: A review of advances in manifold learning.","volume":"31","author":"D.Lunga","year":"2014","journal-title":"Signal Processing Magazine, IEEE"},{"key":"ijncr.2015040103-16","author":"Y.Ma","year":"2012","journal-title":"Manifold Learning Theory and Applications"},{"issue":"4","key":"ijncr.2015040103-17","first-page":"1111","article-title":"Nonlinear dimensionality reduction of data lying on the multicluster manifold. Systems, Man, and Cybernetics, Part B: Cybernetics","volume":"38","author":"D.Meng","year":"2008","journal-title":"IEEE Trans. on"},{"issue":"1","key":"ijncr.2015040103-18","doi-asserted-by":"crossref","first-page":"87","DOI":"10.1137\/S0895479804442334","article-title":"Nonlinear discriminant analysis using kernel functions and the generalized singular value decomposition.","volume":"27","author":"C. H.Park","year":"2005","journal-title":"SIAM Journal on Matrix Analysis and Applications"},{"key":"ijncr.2015040103-19","first-page":"2825","article-title":"Scikit-learn: Machine learning in Python.","volume":"12","author":"F.Pedregosa","year":"2011","journal-title":"Journal of Machine Learning Research"},{"key":"ijncr.2015040103-20","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":"S. T.Roweis","year":"2000","journal-title":"Science"},{"issue":"5","key":"ijncr.2015040103-21","doi-asserted-by":"crossref","first-page":"1299","DOI":"10.1162\/089976698300017467","article-title":"Nonlinear component analysis as a kernel eigenvalue problem.","volume":"10","author":"B.Scholkopf","year":"1998","journal-title":"Neural Computation"},{"key":"ijncr.2015040103-22","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":"J.Tenenbaum","year":"2000","journal-title":"Science"},{"issue":"1","key":"ijncr.2015040103-23","doi-asserted-by":"crossref","first-page":"97","DOI":"10.1007\/BF02127699","article-title":"Barycentric coordinates for convex polytopes.","volume":"6","author":"J. D.Warren","year":"1996","journal-title":"Advances in Computational Mathematics"},{"key":"ijncr.2015040103-24","first-page":"281","article-title":"Manifold learning and applications in recognition","author":"J.Zhang","year":"2004","journal-title":"Intelligent Multimedia Processing with Soft Computing"},{"issue":"1","key":"ijncr.2015040103-25","doi-asserted-by":"crossref","first-page":"133","DOI":"10.1109\/TNNLS.2012.2223825","article-title":"Prime discriminant simplicial complex.","volume":"24","author":"J.Zhang","year":"2013","journal-title":"IEEE Trans. Neural Netw. Learning Syst."}],"container-title":["International Journal of Natural Computing Research"],"original-title":[],"language":"ng","link":[{"URL":"https:\/\/www.igi-global.com\/viewtitle.aspx?TitleId=126482","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,6,1]],"date-time":"2022-06-01T10:43:37Z","timestamp":1654080217000},"score":1,"resource":{"primary":{"URL":"https:\/\/services.igi-global.com\/resolvedoi\/resolve.aspx?doi=10.4018\/ijncr.2015040103"}},"subtitle":[""],"short-title":[],"issued":{"date-parts":[[2015,4,1]]},"references-count":26,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2015,4]]}},"URL":"https:\/\/doi.org\/10.4018\/ijncr.2015040103","relation":{},"ISSN":["1947-928X","1947-9298"],"issn-type":[{"value":"1947-928X","type":"print"},{"value":"1947-9298","type":"electronic"}],"subject":[],"published":{"date-parts":[[2015,4,1]]}}}