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Manifold Learning Dimensionality Reduction

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

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

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Introduction (dim. reduction) Principal Component Analysis x ∑

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

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

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Introduction (dim. reduction)

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

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

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

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Introduction (manifold)

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

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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

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Isomap (Approximating geodesic dist.)

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is not much bigger than

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

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

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

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Isomap (Approximating geodesic dist.)

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

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

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

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

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Isomap

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

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Nonlinear Dimensionality Reduction Approach (ISOMAP)

Nonlinear Dimensionality Reduction Approach (ISOMAP)

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