Exploiting Data Topology in Visualization and Clustering of Self-Organizing Maps

1. Exploiting Data Topology in Visualization and Clustering of Self-Organizing Maps KadimTas ¸demir and ErzsébetMerényi, Senior Member TNN, 2011 Presented by Hung-Yi Cai 2011/3/9

2. Outlines • Motivation • Objectives • Previous Study • Methodology • Experiments • Conclusions • Comments

3. Motivation • Different aspects of the information learned by the SOM are presented by existing methods, but data topology, which is present in the SOM’s knowledge, is greatly underutilized. • Data topology can be integrated into the visualization of the SOM and thereby provide a more elaborate view of the cluster structure than existing schemes.

4. Objectives To integrate the data topology, present in the SOM’s knowledge, into the visualization of the SOM for improved capture of clusters. This objective will be accomplished through a new concept of the “connectivity matrix” and its specific rendering over the SOM.

5. Previous Study • SOM is a topology preserving mapping • Ideally, prototypes(neurons) those are neighbors in SOM map are also neighbors (centroids of neighboring Voronoi polyhedra) in data space and vice versa. • Growing SOM • It appears less robust than the Kohonen SOM because of the large number of parameters needing adjustment. • ViSOM • it requires a relatively large number of prototypes even for small data sets.

6. Methodology • Topology visualization through connectivity matrix of SOM prototypes • CONNvis: visualization of the connectivity matrix • Assessment of topology preservation with CONNvis

7. Topology visualization through connectivity matrix of SOM prototypes • Induced Delaunay Triangulation and Voronoi • It can be determined from the relationships of the best matching units (BMUs) and the second BMUs. • Connectivity Matrix • It is a weighted analog of A, where the weights indicate the density distribution of the input data among the prototypes adjacent in M. • where, RFij means wi is the BMU and wjis the second BMU.

8. CONNvis: visualization of the connectivity matrix • Line width：Global Importance • The strength of the connection and reflects the density distribution among the connected units. • Line colors：Local Importance • A ranking of the connectivity strengths of wi. • Reveals most-to-least dense regions local to wi in data space.

9. The threshold of width

10. Assessment of topology preservation with CONNvis • Topology violations • connected neural units that are not immediate neighbors in map (forward topology violations); • unconnected neural units that are immediate neighbors in map (backward topology violations).

11. Remove weak connections Remove weak connections that link any two coarse clusters X and Y at their boundary

12. Experiments A real remote sensing spectral image of Ocean City

13. Experiments Compare to U-matrix and ISOMAP

14. Conclusions • CONNvis integrates data distribution into the customary Delaunay triangulation, which, when displayed on the SOM grid, enables 2-D visualization of the manifold structure regardless of the data dimensionality. • CONNvis is also unique among SOM representations in that it shows both forward and backward topology violations on the SOM grid.

15. Comments • Advantages • CONNvis greatly assists in detailed identification of cluster boundaries. • Applications • Data Clustering