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Extending metric multidimensional scaling with Bregman divergences

Extending metric multidimensional scaling with Bregman divergences. Mr. Jigang Sun Supervisor: Prof. Colin Fyfe Nov 2009. Multidimensional Scaling(MDS).

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Extending metric multidimensional scaling with Bregman divergences

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  1. Extending metric multidimensional scaling with Bregman divergences Mr. Jigang Sun Supervisor: Prof. Colin Fyfe Nov 2009

  2. Multidimensional Scaling(MDS) • A group of information visualisation methods that projects data from high dimensional space, to a low dimensional space, often two or three dimensions, keeping inter-point dissimilarities (e.g. distances) in low dimensional space as close as possible to the original dissimilarities in high dimensional space. When Euclidean distances are used, it is Metric MDS.

  3. An example low dimensional space /latent space/output space high dimensional space /data space/input space basic MDS

  4. Basic MDS • We minimise the stress function data space Latent space

  5. Sammon Mapping (1969) Focuses on small distances: for the same error, the smaller distance is given bigger stress, thus on average the small distances are mapped more accurately than long distances. Small neighbourhoods are well preserved.

  6. Bregman divergence is the Bregman divergence between p and q based on strictly convex function, F. Intuitively, the difference between the value of F at point p and the value of the first-order Taylor expansion of F around point q evaluated at point p.

  7. Bregman divergence • When F is in one variable, the Bregman Divergence is truncated Taylor series • A useful property for MDS: Non-negativity: • If is a function in p, p approaches q when it is minimised.

  8. MDS using Bregman divergence • Bregmanised MDS • Equivalent Expression: residual Taylor series

  9. Basic MDS is a special BMMDS • Base convex function is chosen as • And higher order derivatives are • So • Is derived as

  10. Example 2: Extended Sammon • Base convex function • This is equivalent to • The Sammon mapping is rewritten as

  11. Sammon and Extended Sammon • The common term • The Sammon mapping is considered to be an approximation to the Extended Sammon mapping using the common term. • The Extended Sammon mapping will do more adjustments on the basis of the higher order terms.

  12. An Experiment on Swiss roll data set

  13. At a glance • Basic MDS captures the global curve, but poorly differentiates local points of same X and Y coordinate but different Z coordinate. • The Sammon mapping does better than BasicMDS. • The Extended Sammon mapping is the best.

  14. Distance preservation

  15. Distance preservation • Horizontal axis: mean distances in data space, 40 sets. • Vertical axis: relative mean distances in latent space. • Sammon is better than BasicMDS, Extended Sammon is better than Sammon: • Small distances are mapped closer to their original value in data space; long distances are mapped longer.

  16. Relative standard deviation

  17. Relative standard deviation • On short distances, Sammon has smaller variance than BasicMDS, Extended Sammon has smaller variance than Sammon, i.e. control of small distances is enhanced. • Large distances are given more and more freedom in the same order as above.

  18. LCMC: local continuity meta-criterion (L. Chen 2006) • A common measure assesses projection quality of different MDS methods. • In terms of neighbourhood preservation. • Value between 0 and 1, the higher the better.

  19. Quality accessed by LCMC

  20. Stress comparison between Sammon and Extended Sammon

  21. Stress comparison between Sammon and Extended Sammon • For the ExtendedSammon, a shorter distance error (e.g. if Dij-Lij=2) in latent space is penalized more than a longer distance error (e.g. if Dij – Lij =-2)in latent space.

  22. Stress formation by items

  23. Stress formation by terms • Stress coming from the term of the Sammon mapping is the largest. It is the main part of stress. • However, for small distances, the contribution from other terms is not negligible.

  24. OpenBox, Sammon and FirstGroup

  25. SecondGroup on OpenBox

  26. Future work • Combining two opposite strategies for choosing base convex functions. • Right Bregman divergences is one kind of CCA.

  27. Conclusion • Applied Bregman divergences to multidimensional scaling. • Shown that basic MMDS is a special case and Sammon mapping approximates a BMMDS. • Improved upon both with 2 families of divergences. • Shown results on two artificial data sets.

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