Neurocomputing 57 (2004) 49–76 www.elsevier.com/locate/neucom Nonlinear projection with curvilinear distances: Isomap versus curvilinear distance analysis John Aldo Lee a ; ∗ , Amaury Lendasse b , Michel Verleysen a ; 1 a Microelectronics Laboratory, Department of Electricity, Universit e catholique de Louvain, Place du Levant, 3, B-1348 Louvain-la-Neuve, Belgium b Universit e catholique de Louvain, CESAME, Avenue George Lema ˆ itre, 4, B-1348 Louvain-la-Neuve, Belgium Abstract Dimension reduction techniques are widely used for the analysis and visualization of complex sets of data. This paper compares two recently published methods for nonlinear projection: Isomap and Curvilinear Distance Analysis (CDA). Contrarily to the traditional linear PCA, these methods work like multidimensional scaling, by reproducing in the projection space the pairwise distances measured in the data space. However, they dier from the classical linear MDS by the metrics they use and by the way they build the mapping (algebraic or neural). While Isomap relies directly on the traditional MDS, CDA is based on a nonlinear variant of MDS, called Curvilinear Component Analysis (CCA). Although Isomap and CDA share the same metric, the comparison highlights their respective strengths and weaknesses. c  2004 Elsevier B.V. All rights reserved. Keywords: Nonlinear projection; Nonlinear dimensionality reduction; Geodesic distance; Curvilinear distance 1. Introduction When analyzing huge sets of numerical data, problems often occur when the raw data are high-dimensional. For example, these diculties are typical in domains like image processing (large number of pixels) or biomedical signal analysis (numerous captors). * Corresponding author. Tel.: +32-2-47-81-33; fax: +32-2-47-25-98. E-mail addresses: lee@dice.ucl.ac.be (J.A. Lee), lendasse@auto.ucl.ac.be (A. Lendasse), verleysen@dice.ucl.ac.be (M. Verleysen). 1 M.V. works as a senior research associate of the Belgian FNRS. 0925-2312/$-see front matter c  2004 Elsevier B.V. All rights reserved. doi:10.1016/j.neucom.2004.01.007