1. Manifold Learning Dimensionality Reduction

2. Outline Introduction Dim. Reduction Manifold Isomap Overall procedure Approximating geodesic dist. Dijkstra’s algorithm Reference

3. Introduction (dim. reduction) Dimensionality Reduction Linear PCA MDS Non-linear Isomap(2000) LLE(2000) SDE(2005)

4. Introduction (dim. reduction) Principal Component Analysis x ∑

5. Introduction (dim. reduction) Dimensionality Reduction Linear PCA MDS Non-linear Isomap(2000) LLE(2000) SDE(2005)

6. Introduction (dim. reduction) Multidimensional Scaling ChicagoRaleighBostonSeattleS.F.AustinOrlando Chicago0 Raleigh6410 Boston8516080 Seattle1733236324880 S.F.1855240626966840 Austin97211671691176414950 Orlando99452011052565245810150

7. Introduction (dim. reduction)

8. Dimensionality Reduction Linear PCA MDS Non-linear Isomap(2000) LLE(2000) SDE(2005)

9. Introduction (manifold) Linear methods do nothing more than “ globally transform ” (rotate/translate..) data. Sometimes need to “ unwrap ” the data first PCA

10. Introduction (dim. reduction) The task of dimensionality reduction is to find a small number of features to represent a large number of observed dimensions.

11. Introduction (manifold)

13. Outline Introduction Dim. Reduction Manifold Isomap Overall procedure Approximating geodesic dist. Dijkstra’s algorithm Reference

14. Isomap (overall procedure) Compute fully-connected neighborhood of points for each item (k nearest) Calculate pairwise Euclidean distances within each neighborhood Use Dijkstra ’ s Algorithm to compute shortest path from each point to non- neighboring points Run MDS on resulting distance matrix

15. Isomap (Approximating geodesic dist.)

17. is not much bigger than

18. Isomap (Approximating geodesic dist.) is not much bigger than

19. Isomap (Approximating geodesic dist.) is not much bigger than

20. Isomap (Approximating geodesic dist.) is not much bigger than

21. Isomap (Approximating geodesic dist.)

22. Isomap (Dijkstra ’ s Algorithm) Greedy breadth-first algorithm to compute shortest path from one point to all other points

23. Isomap (Dijkstra ’ s Algorithm) Greedy breadth-first algorithm to compute shortest path from one point to all other points

24. Isomap (Dijkstra ’ s Algorithm) Greedy breadth-first algorithm to compute shortest path from one point to all other points

25. Isomap (Dijkstra ’ s Algorithm) Greedy breadth-first algorithm to compute shortest path from one point to all other points

26. Isomap

28. Reference http://www.cs.unc.edu/Courses/comp290-090-s06/ http://www.cse.msu.edu/~lawhiu/manifold/