Manifold Learning Dimensionality Reduction. Outline Introduction Dim. Reduction Manifold Isomap...

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Manifold Learning Dimensionality Reduction

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Transcript of Manifold Learning Dimensionality Reduction. Outline Introduction Dim. Reduction Manifold Isomap...

Page 1: Manifold Learning Dimensionality Reduction. Outline Introduction Dim. Reduction Manifold Isomap Overall procedure Approximating geodesic dist. Dijkstra’s.

Manifold LearningDimensionality Reduction

Page 2: Manifold Learning Dimensionality Reduction. Outline Introduction Dim. Reduction Manifold Isomap Overall procedure Approximating geodesic dist. Dijkstra’s.

Outline

Introduction Dim. Reduction Manifold

Isomap Overall procedure Approximating geodesic dist. Dijkstra’s algorithm

Reference

Page 3: Manifold Learning Dimensionality Reduction. Outline Introduction Dim. Reduction Manifold Isomap Overall procedure Approximating geodesic dist. Dijkstra’s.

Introduction (dim. reduction)

DimensionalityReduction

LinearPCAMDS

Non-linearIsomap(2000)

LLE(2000)SDE(2005)

Page 4: Manifold Learning Dimensionality Reduction. Outline Introduction Dim. Reduction Manifold Isomap Overall procedure Approximating geodesic dist. Dijkstra’s.

Introduction (dim. reduction)

Principal Component Analysis

x∑

Page 5: Manifold Learning Dimensionality Reduction. Outline Introduction Dim. Reduction Manifold Isomap Overall procedure Approximating geodesic dist. Dijkstra’s.

Introduction (dim. reduction)

DimensionalityReduction

LinearPCAMDS

Non-linearIsomap(2000)

LLE(2000)SDE(2005)

Page 6: Manifold Learning Dimensionality Reduction. Outline Introduction Dim. Reduction Manifold Isomap Overall procedure Approximating geodesic dist. Dijkstra’s.

Introduction (dim. reduction)

Multidimensional Scaling

ChicagoRaleigh

Boston Seattle S.F. Austin Orlando

Chicago 0

Raleigh 641 0

Boston 851 608 0

Seattle 1733 2363 2488 0

S.F. 1855 2406 2696 684 0

Austin 972 1167 1691 1764 1495 0

Orlando 994 520 1105 2565 2458 1015 0

Page 7: Manifold Learning Dimensionality Reduction. Outline Introduction Dim. Reduction Manifold Isomap Overall procedure Approximating geodesic dist. Dijkstra’s.

Introduction (dim. reduction)

Page 8: Manifold Learning Dimensionality Reduction. Outline Introduction Dim. Reduction Manifold Isomap Overall procedure Approximating geodesic dist. Dijkstra’s.

Introduction (dim. reduction)

DimensionalityReduction

LinearPCAMDS

Non-linearIsomap(2000)

LLE(2000)SDE(2005)

Page 9: Manifold Learning Dimensionality Reduction. Outline Introduction Dim. Reduction Manifold Isomap Overall procedure Approximating geodesic dist. Dijkstra’s.

Introduction (manifold)

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

PCA

Page 10: Manifold Learning Dimensionality Reduction. Outline Introduction Dim. Reduction Manifold Isomap Overall procedure Approximating geodesic dist. Dijkstra’s.

Introduction (dim. reduction)

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

Page 11: Manifold Learning Dimensionality Reduction. Outline Introduction Dim. Reduction Manifold Isomap Overall procedure Approximating geodesic dist. Dijkstra’s.

Introduction (manifold)

Page 12: Manifold Learning Dimensionality Reduction. Outline Introduction Dim. Reduction Manifold Isomap Overall procedure Approximating geodesic dist. Dijkstra’s.

Introduction (manifold)

Page 13: Manifold Learning Dimensionality Reduction. Outline Introduction Dim. Reduction Manifold Isomap Overall procedure Approximating geodesic dist. Dijkstra’s.

Outline

Introduction Dim. Reduction Manifold

Isomap Overall procedure Approximating geodesic dist. Dijkstra’s algorithm

Reference

Page 14: Manifold Learning Dimensionality Reduction. Outline Introduction Dim. Reduction Manifold Isomap Overall procedure Approximating geodesic dist. Dijkstra’s.

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

Page 15: Manifold Learning Dimensionality Reduction. Outline Introduction Dim. Reduction Manifold Isomap Overall procedure Approximating geodesic dist. Dijkstra’s.

Isomap (Approximating geodesic dist.)

Page 16: Manifold Learning Dimensionality Reduction. Outline Introduction Dim. Reduction Manifold Isomap Overall procedure Approximating geodesic dist. Dijkstra’s.

Isomap (Approximating geodesic dist.)

Page 17: Manifold Learning Dimensionality Reduction. Outline Introduction Dim. Reduction Manifold Isomap Overall procedure Approximating geodesic dist. Dijkstra’s.

Isomap (Approximating geodesic dist.)

is not much bigger than

Page 18: Manifold Learning Dimensionality Reduction. Outline Introduction Dim. Reduction Manifold Isomap Overall procedure Approximating geodesic dist. Dijkstra’s.

Isomap (Approximating geodesic dist.)

is not much bigger than

Page 19: Manifold Learning Dimensionality Reduction. Outline Introduction Dim. Reduction Manifold Isomap Overall procedure Approximating geodesic dist. Dijkstra’s.

Isomap (Approximating geodesic dist.)

is not much bigger than

Page 20: Manifold Learning Dimensionality Reduction. Outline Introduction Dim. Reduction Manifold Isomap Overall procedure Approximating geodesic dist. Dijkstra’s.

Isomap (Approximating geodesic dist.)

is not much bigger than

Page 21: Manifold Learning Dimensionality Reduction. Outline Introduction Dim. Reduction Manifold Isomap Overall procedure Approximating geodesic dist. Dijkstra’s.

Isomap (Approximating geodesic dist.)

Page 22: Manifold Learning Dimensionality Reduction. Outline Introduction Dim. Reduction Manifold Isomap Overall procedure Approximating geodesic dist. Dijkstra’s.

Isomap (Dijkstra’s Algorithm)

Greedy breadth-first algorithm to compute shortest path from one point to all other points

Page 23: Manifold Learning Dimensionality Reduction. Outline Introduction Dim. Reduction Manifold Isomap Overall procedure Approximating geodesic dist. Dijkstra’s.

Isomap (Dijkstra’s Algorithm)

Greedy breadth-first algorithm to compute shortest path from one point to all other points

Page 24: Manifold Learning Dimensionality Reduction. Outline Introduction Dim. Reduction Manifold Isomap Overall procedure Approximating geodesic dist. Dijkstra’s.

Isomap (Dijkstra’s Algorithm)

Greedy breadth-first algorithm to compute shortest path from one point to all other points

Page 25: Manifold Learning Dimensionality Reduction. Outline Introduction Dim. Reduction Manifold Isomap Overall procedure Approximating geodesic dist. Dijkstra’s.

Isomap (Dijkstra’s Algorithm)

Greedy breadth-first algorithm to compute shortest path from one point to all other points

Page 26: Manifold Learning Dimensionality Reduction. Outline Introduction Dim. Reduction Manifold Isomap Overall procedure Approximating geodesic dist. Dijkstra’s.

Isomap

Page 27: Manifold Learning Dimensionality Reduction. Outline Introduction Dim. Reduction Manifold Isomap Overall procedure Approximating geodesic dist. Dijkstra’s.

Isomap

Page 28: Manifold Learning Dimensionality Reduction. Outline Introduction Dim. Reduction Manifold Isomap Overall procedure Approximating geodesic dist. Dijkstra’s.

Reference

http://www.cs.unc.edu/Courses/comp290-090-s06/

http://www.cse.msu.edu/~lawhiu/manifold/


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