Nonlinear Dimensionality Reduction Frameworks

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# Nonlinear Dimensionality Reduction Frameworks - PowerPoint PPT Presentation

Nonlinear Dimensionality Reduction Frameworks. Rong Xu Chan su Lee. Outline. Intuition of Nonlinear Dimensionality Reduction(NLDR) ISOMAP LLE NLDR in Gait Analysis. Intuition: how does your brain store these pictures?. Brain Representation. Brain Representation. Every pixel?

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### Nonlinear Dimensionality Reduction Frameworks

Rong Xu

Chan su Lee

Outline
• Intuition of Nonlinear Dimensionality Reduction(NLDR)
• ISOMAP
• LLE
• NLDR in Gait Analysis
Brain Representation
• Every pixel?
• Or perceptually meaningful structure?
• Up-down pose
• Left-right pose
• Lighting direction

So, your brain successfully reduced the high-dimensional inputs to an intrinsically 3-dimensional manifold!

Manifold Learning
• A manifold is a topological space which is locally Euclidean
• An example of nonlinear manifold:
Discover low dimensional representations (smooth manifold) for data in high dimension.

Linear approaches(PCA, MDS) vs Non-linear approaches (ISOMAP, LLE)

Manifold Learning

latent

Y

X

observed

Linear Approach- PCA
• PCA Finds subspace linear projections of input data.
Linear Approach- MDS
• MDS takes a matrix of pairwise distances and gives a mapping to Rd. It finds an embedding that preserves the interpoint distances, equivalent to PCA when those distance are Euclidean.
• BUT! PCA and MDS both fail to do embedding with nonlinear data, like swiss roll.
Constructing neighbourhood graph G

For each pair of points in G, Computing shortest path distances ---- geodesic distances.

Use Classical MDS with geodesic distances.

Euclidean distance Geodesic distance

Nonlinear Approaches- ISOMAP

Josh. Tenenbaum, Vin de Silva, John langford 2000

Sample points with Swiss Roll
• Altogether there are 20,000 points in the “Swiss roll” data set. We sample 1000 out of 20,000.
Construct neighborhood graph G

K- nearest neighborhood (K=7)

DG is 1000 by 1000 (Euclidean) distance matrix of two neighbors (figure A)

Compute all-points shortest path in G

Now DG is 1000 by 1000 geodesic distance matrix of two arbitrary points along the manifold(figure B)

Use MDS to embed graph in Rd
• Find a d-dimensional Euclidean space Y(Figure c) to minimize the cost function:
Linear Approach-classical MDS
• Theorem: For any squared distance matrix ,there exists of points xi and,xj, such that
• So
Solution
• Y lies in Rd and consists of N points correspondent to the N original points in input space.