A nonlinear projection method based on Kohonen's topology preserving maps

A nonlinear projection method is presented to visualize high-dimensional data as a 2D image. The proposed method is based on the topology preserving mapping algorithm of Kohonen. The topology preserving mapping algorithm is used to train a 2D network structure. Then the interpoint distances in the feature space between the units in the network are graphically displayed to show the underlying structure of the data. Furthermore, we present and discuss a new method to quantify how well a topology preserving mapping algorithm maps the high-dimensional input data onto the network structure. This is used to compare our projection method with a well-known method of Sammon (1969). Experiments indicate that the performance of the Kohonen projection method is comparable or better than Sammon's method for the purpose of classifying clustered data. Its time-complexity only depends on the resolution of the output image, and not on the size of the dataset. A disadvantage, however, is the large amount of CPU time required.

[1]  Anil K. Jain,et al.  Unsupervised texture segmentation using Gabor filters , 1990, 1990 IEEE International Conference on Systems, Man, and Cybernetics Conference Proceedings.

[2]  R. Fisher.  THE USE OF MULTIPLE MEASUREMENTS IN TAXONOMIC PROBLEMS , 1936 .

[3]  Klaus Schulten,et al.  Topology-conserving maps for learning visuo-motor-coordination , 1989, Neural Networks.

[4]  Anil K. Jain,et al.  Algorithms for Clustering Data , 1988 .

[5]  C. E. Pykett.  Improving the efficiency of Sammon's nonlinear mapping by using clustering archetypes , 1978 .

[6]  Keinosuke Fukunaga,et al.  A Nonparametric Two-Dimensional Display for Classification , 1982, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[7]  John W. Tukey,et al.  A Projection Pursuit Algorithm for Exploratory Data Analysis , 1974, IEEE Transactions on Computers.

[8]  Teuvo Kohonen,et al.  The self-organizing map , 1990 .

[9]  Gautam Biswas,et al.  Evaluation of Projection Algorithms , 1981, IEEE Transactions on Pattern Analysis and Machine Intelligence.

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

[11]  Bruce J. Schachter.  A nonlinear mapping algorithm for large data sets , 1978 .

[12]  J. MacQueen.  Some methods for classification and analysis of multivariate observations , 1967 .

[13]  Richard C. T. Lee,et al.  A Heuristic Relaxation Method for Nonlinear Mapping in Cluster Analysis , 1973, IEEE Trans. Syst. Man Cybern..

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

[15]  Anil K. Jain,et al.  Texture classification and segmentation using multiresolution simultaneous autoregressive models , 1992, Pattern Recognit..

[16]  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.

[17]  Teuvo Kohonen,et al.  The 'neural' phonetic typewriter , 1988, Computer.

[18]  Teuvo Kohonen,et al.  Self-Organization and Associative Memory , 1988 .

[19]  Dakai Wang,et al.  The equal-angle spanning tree mapping: A sequential method for projecting from h-space to 2-space , 1983, Pattern Recognit. Lett..

[20]  I. White.  Comment on "A Nonlinear Mapping for Data Structure Analysis" , 1972, IEEE Trans. Computers.

[21]  C. D. Gelatt,et al.  Optimization by Simulated Annealing , 1983, Science.

[22]  Piet W. Verbeek,et al.  Shading from shape, the eikonal equation solved by grey-weighted distance transform , 1990, Pattern Recognit. Lett..

[23]  Richard C. Dubes,et al.  Experiments in projection and clustering by simulated annealing , 1989, Pattern Recognit..