Curvilinear Component Analysis for High-Dimensional Data Representation: I. Theoretical Aspects and Practical Use in the Presence of Noise

Starting from a recall of the theoretical framework, this paper presents the conditions and the strategy of implementation of CCA, a recent algorithm for non-linear mapping. Initially developed in a basic form, for non-linear and high-dimensional data sets, the algorithm is here adapted to the general, and more realistic, case of noisy data. This algorithm, which finds the manifold (in particular, the intrinsic dimension) of the data, has proved to be very efficient in the representation of highly folded data structures. We describe here how it can be tuned to find the average manifold and how robust the convergence is. A companion paper (this issue) presents various applications using this property.

[1]  J. Kruskal.  Nonmetric multidimensional scaling: A numerical method , 1964 .

[2]  Jack Sklansky,et al.  An overview of mapping techniques for exploratory pattern analysis , 1988, Pattern Recognit..

[3]  Maurizio Cirrincione,et al.  Diagnosis of Three-Phase Converters using the VQP Neural Network , 1994 .

[4]  P. Groenen,et al.  Modern Multidimensional Scaling: Theory and Applications , 1999 .

[5]  Jean-Luc Schwartz,et al.  Models for Audiovisual Fusion in a Noisy-Vowel Recognition Task , 1998, J. VLSI Signal Process..

[6]  John W. Sammon,et al.  A Nonlinear Mapping for Data Structure Analysis , 1969, IEEE Transactions on Computers.

[7]  R. Shepard.  The analysis of proximities: Multidimensional scaling with an unknown distance function. II , 1962 .

[8]  Teuvo Kohonen,et al.  Self-Organization and Associative Memory, Third Edition , 1989, Springer Series in Information Sciences.

[9]  Pierre Demartines.  Analyse de donnees par reseaux de neurones auto-organises , 1994 .

[10]  Jeanny Hérault,et al.  Curvilinear component analysis: a self-organizing neural network for nonlinear mapping of data sets , 1997, IEEE Trans. Neural Networks.

[11]  Allen Gersho,et al.  Vector quantization and signal compression , 1991, The Kluwer international series in engineering and computer science.

[12]  Anne Guérin-Dugué,et al.  Curvilinear Component Analysis for High-Dimensional Data Representation: II. Examples of Additional Mapping Constraints in Specific Applications , 1999, IWANN.