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Manifold Learning Dimensionality Reduction. Outline Introduction Dim. Reduction Manifold Isomap Overall procedure Approximating geodesic dist. Dijkstra’s.

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Presentation on theme: "Manifold Learning Dimensionality Reduction. Outline Introduction Dim. Reduction Manifold Isomap Overall procedure Approximating geodesic dist. Dijkstra’s."— Presentation transcript:

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 Boston Seattle S.F Austin Orlando

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)

12

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.)

16

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

27

28 Reference


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