author = "Miranda Junior, Gastao Florencio and Thomaz, Carlos Eduardo and 
                         Giraldi, Gilson Antonio",
          affiliation = "Department of Mathematics, Federal University of Sergipe, Aracaju, 
                         Brazil and Department of Electrical Engineering, FEI, Sao Bernardo 
                         do Campo, Brazil and Department of Mathematics and Computational 
                         Methods, National Laboratory for Scientific Computing, Petropolis, 
                title = "Geometric Data Analysis Based on Manifold Learning with 
                         Applications for Image Understanding",
            booktitle = "Proceedings...",
                 year = "2017",
               editor = "Torchelsen, Rafael Piccin and Nascimento, Erickson Rangel do and 
                         Panozzo, Daniele and Liu, Zicheng and Farias, Myl{\`e}ne and 
                         Viera, Thales and Sacht, Leonardo and Ferreira, Nivan and Comba, 
                         Jo{\~a}o Luiz Dihl and Hirata, Nina and Schiavon Porto, Marcelo 
                         and Vital, Creto and Pagot, Christian Azambuja and Petronetto, 
                         Fabiano and Clua, Esteban and Cardeal, Fl{\'a}vio",
         organization = "Conference on Graphics, Patterns and Images, 30. (SIBGRAPI)",
            publisher = "Sociedade Brasileira de Computa{\c{c}}{\~a}o",
              address = "Porto Alegre",
             keywords = "manifold learning, statistical learning, Riemannian manifolds, 
                         image analysis, deep learning.",
             abstract = "Nowadays, pattern recognition, computer vision, signal processing 
                         and medical image analysis, require the managing of large amount 
                         of multidimensional image databases, possibly sampled from 
                         nonlinear manifolds. The complex tasks involved in the analysis of 
                         such massive data lead to a strong demand for nonlinear methods 
                         for dimensionality reduction to achieve efficient representation 
                         for information extraction. In this avenue, manifold learning has 
                         been applied to embed nonlinear image data in lower dimensional 
                         spaces for subsequent analysis. The result allows a geometric 
                         interpretation of image spaces with relevant consequences for data 
                         topology, computation of image similarity, discriminant 
                         analysis/classification tasks and, more recently, for deep 
                         learning issues. In this paper, we firstly review Riemannian 
                         manifolds that compose the mathematical background in this field. 
                         Such background offers the support to set up a data model that 
                         embeds usual linear subspace learning and discriminant analysis 
                         results in local structures built from samples drawn from some 
                         unknown distribution. Afterwards, we discuss topological issues in 
                         data preparation for manifold learning algorithms as well as the 
                         determination of manifold dimension. Then, we survey 
                         dimensionality reduction techniques with particular attention to 
                         Riemannian manifold learning. Besides, we discuss the application 
                         of concepts in discrete and polyhedral geometry for synthesis and 
                         data clustering over the recovered Riemannian manifold with 
                         emphasis in face images in the computational experiments. Next, we 
                         discuss promising perspectives of manifold learning and related 
                         topics for image analysis, classification and relationships with 
                         deep learning methods. Specifically, we discuss the application of 
                         foliation theory, discriminant analysis and kernel methods in 
                         curved spaces. Besides, we take differential geometry in manifolds 
                         as a paradigm to discuss deep generative models and metric 
                         learning algorithms.",
  conference-location = "Niter{\'o}i, RJ",
      conference-year = "Oct. 17-20, 2017",
             language = "en",
                  ibi = "8JMKD3MGPAW/3PJRSSH",
                  url = "",
           targetfile = "PID4980343.pdf",
        urlaccessdate = "2021, Mar. 02"