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Curvilinear Component Analysis and Bregman divergences

Explore multidimensional scaling and visualization methods to project high-dimensional data into a low-dimensional space while preserving inter-point dissimilarities. Improve Sammon Mapping with Bregman divergences. Experiment with Swiss roll data set and compare different convex functions. Introduce Curvilinear Component Analysis with Bregman divergences.

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Curvilinear Component Analysis and Bregman divergences

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  1. Curvilinear Component Analysis and Bregman divergences Jigang Sun Colin Fyfe Malcolm Crowe 28 April 2010 University of the West of Scotland

  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. Visualising 18 dimensional data

  4. Basic MDS • The basic MDS, the stress function to be minimised • Sammon Mapping (1969) • Improve the Sammon mapping with Bregman divergence

  5. Bregman divergence Intuitively, it is 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. q p

  6. 2 representations • When F is in one variable, the Bregman Divergence is truncated Taylor series • Two useful properties for MDS 1. Non-negativity 2. Non-symmetry Except in special cases such as F(x)=x^2

  7. Improving Sammon Mapping with Bregman divergences • Recall the classical Sammon Mapping (1969) • Choose a base convex function • common term: the first term of ExtendedSammon is Sammon, not considering constant coefficients

  8. An Experiment on Swiss roll data set

  9. Two groups of Convex functions • No 1 is for the Extended Sammon mapping.

  10. OpenBox, Sammon and FirstGroup

  11. SecondGroup on OpenBox

  12. Curvilinear Component Analysis (CCA) and Bregman Divergences • W( .) has argument the inter-point distance in latent space • Good at unfolding strongly nonlinear structures • Stochastic gradient descent updating rule

  13. A version of CCA One weight function can be Updating rule

  14. Rewriting stress function for CCA using right Bregman divergences Given convex function Updating rule is the same

  15. The common term between BasicCCA and Real CCA = • The first term is common with

  16. Real CCA vs Basic CCA

  17. Conclusions • We introduced • The Extended Sammon mapping vs the Sammon mapping • We create two groups of left Bregman divergences and experiment on artificial data sets. • A right Bregman divergence redefines the stress function for Curvilinear Component Analysis • Any questions?

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