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

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Presentation transcript:

Manifold Learning Dimensionality Reduction

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

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

Introduction (dim. reduction) Principal Component Analysis x ∑

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

Introduction (dim. reduction) Multidimensional Scaling ChicagoRaleighBostonSeattleS.F.AustinOrlando Chicago0 Raleigh6410 Boston Seattle S.F Austin Orlando

Introduction (dim. reduction)

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

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

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

Introduction (manifold)

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

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

Isomap (Approximating geodesic dist.)

is not much bigger than

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

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

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

Isomap (Approximating geodesic dist.)

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

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

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

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

Isomap

Reference