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Nonlinear Dimensionality Reduction Frameworks

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

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  1. Nonlinear Dimensionality Reduction Frameworks Rong Xu Chan su Lee

  2. Outline • Intuition of Nonlinear Dimensionality Reduction(NLDR) • ISOMAP • LLE • NLDR in Gait Analysis

  3. Intuition: how does your brain store these pictures?

  4. Brain Representation

  5. 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!

  6. Data for Faces

  7. Data for Handwritten “2”

  8. Data for Hands

  9. Manifold Learning • A manifold is a topological space which is locally Euclidean • An example of nonlinear manifold:

  10. 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

  11. Linear Approach- PCA • PCA Finds subspace linear projections of input data.

  12. 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.

  13. 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

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

  15. Construct neighborhood graph G K- nearest neighborhood (K=7) DG is 1000 by 1000 (Euclidean) distance matrix of two neighbors (figure A)

  16. 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)

  17. Use MDS to embed graph in Rd • Find a d-dimensional Euclidean space Y(Figure c) to minimize the cost function:

  18. Linear Approach-classical MDS • Theorem: For any squared distance matrix ,there exists of points xi and,xj, such that • So

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

  20. PCA, MD vs ISOMAP

  21. Isomap: Advantages • 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.

  22. Isomap: Disadvantages • 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.

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