Extending Sammon mapping with Bregman divergences

The Sammon mapping has been one of the most successful nonlinear metric multidimensional scaling methods since its advent in 1969, but effort has been focused on algorithm improvement rather than on the form of the stress function. This paper further investigates using left Bregman divergences to extend the Sammon mapping and by analogy develops right Bregman divergences and reveals the mechanism that improves the performance of scaling over the Sammon mapping. The influence of data space distance preprocessing on optimisation speed is noticed. Non-stress visualisation quality measures are used to compare the configuration quality of the Sammon mapping and its extensions using both Euclidean distance and graph distance on three data sets.

[1]  J. Tenenbaum,et al.  A global geometric framework for nonlinear dimensionality reduction. , 2000, Science.

[2]  Jeanny Hérault,et al.  Curvilinear Component Analysis for High-Dimensional Data Representation: I. Theoretical Aspects and Practical Use in the Presence of Noise , 1999, IWANN.

[3]  Anil K. Jain,et al.  Artificial neural networks for feature extraction and multivariate data projection , 1995, IEEE Trans. Neural Networks.

[4]  Colin Fyfe,et al.  Curvilinear component analysis and Bregman divergences , 2010, ESANN.

[5]  W. Torgerson.  Multidimensional scaling: I. Theory and method , 1952 .

[6]  Jarkko Venna,et al.  Local multidimensional scaling , 2006, Neural Networks.

[7]  J. Kruskal.  Multidimensional scaling by optimizing goodness of fit to a nonmetric hypothesis , 1964 .

[8]  Michel Verleysen,et al.  Quality assessment of dimensionality reduction: Rank-based criteria , 2009, Neurocomputing.

[9]  Inderjit S. Dhillon,et al.  Clustering with Bregman Divergences , 2005, J. Mach. Learn. Res..

[10]  Li Yang.  Sammon's nonlinear mapping using geodesic distances , 2004, ICPR 2004.

[11]  Michel Verleysen,et al.  Curvilinear Distance Analysis versus Isomap , 2002, ESANN.

[12]  Aynur A. Dayanik.  Feature interval learning algorithms for classification , 2010, Knowl. Based Syst..

[13]  Jun Guo,et al.  Learning a locality discriminating projection for classification , 2009, Knowl. Based Syst..

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

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

[16]  A. Buja,et al.  Local Multidimensional Scaling for Nonlinear Dimension Reduction, Graph Drawing, and Proximity Analysis , 2009 .

[17]  P. Groenen,et al.  Modern multidimensional scaling , 1996 .

[18]  Richard C. T. Lee,et al.  A Triangulation Method for the Sequential Mapping of Points from N-Space to Two-Space , 1977, IEEE Transactions on Computers.

[19]  Colin Fyfe,et al.  Extending metric multidimensional scaling with Bregman divergences , 2011, Pattern Recognit..