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Manifold LearningDimensionality Reduction
Outline
Introduction Dim. Reduction Manifold
Isomap Overall procedure Approximating geodesic dist. Dijkstra’s algorithm
Reference
Introduction (dim. reduction)
DimensionalityReduction
LinearPCAMDS
Non-linearIsomap(2000)
LLE(2000)SDE(2005)
Introduction (dim. reduction)
Principal Component Analysis
x∑
Introduction (dim. reduction)
DimensionalityReduction
LinearPCAMDS
Non-linearIsomap(2000)
LLE(2000)SDE(2005)
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
Introduction (dim. reduction)
Introduction (dim. reduction)
DimensionalityReduction
LinearPCAMDS
Non-linearIsomap(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)
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.)
Isomap (Approximating geodesic dist.)
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
Isomap
Reference
http://www.cs.unc.edu/Courses/comp290-090-s06/
http://www.cse.msu.edu/~lawhiu/manifold/