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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

• Intuition of Nonlinear Dimensionality Reduction(NLDR)

• ISOMAP

• LLE

• NLDR in Gait Analysis

• 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!

• 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 for data in high dimension.

• PCA Finds subspace linear projections of input data.

Linear Approach- MDS for data in high dimension.

• 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 data in high dimension.

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 for data in high dimension.

• Altogether there are 20,000 points in the “Swiss roll” data set. We sample 1000 out of 20,000.

Construct neighborhood graph G for data in high dimension.

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 for data in high dimension.

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

Use MDS to embed graph in R for data in high dimension.d

• Find a d-dimensional Euclidean space Y(Figure c) to minimize the cost function:

Linear Approach-classical MDS for data in high dimension.

• Theorem: For any squared distance matrix ,there exists of points xi and,xj, such that

• So

Solution for data in high dimension.

• Y lies in Rd and consists of N points correspondent to the N original points in input space.

PCA, MD vs ISOMAP for data in high dimension.

Isomap: Advantages for data in high dimension.

• Nonlinear

• Globally optimal

• Still produces globally optimal low-dimensional Euclidean representation even though input space is highly folded, twisted, or curved.

• Guarantee asymptotically to recover the true dimensionality.

Isomap: Disadvantages for data in high dimension.

• May not be stable, dependent on topology of data

• Guaranteed asymptotically to recover geometric structure of nonlinear manifolds

• As N increases, pairwise distances provide better approximations to geodesics, but cost more computation

• If N is small, geodesic distances will be very inaccurate.