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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. Intuition: how does your brain store these pictures?. Brain Representation. Brain Representation. Every pixel?

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Presentation Transcript
outline
Outline
  • Intuition of Nonlinear Dimensionality Reduction(NLDR)
  • ISOMAP
  • LLE
  • NLDR in Gait Analysis
brain representation5
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
Manifold Learning
  • A manifold is a topological space which is locally Euclidean
  • An example of nonlinear manifold:
manifold learning10
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
Linear Approach- PCA
  • PCA Finds subspace linear projections of input data.
linear approach mds
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.
nonlinear approaches isomap
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
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
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
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 r d
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
Linear Approach-classical MDS
  • Theorem: For any squared distance matrix ,there exists of points xi and,xj, such that
  • So
solution
Solution
  • Y lies in Rd and consists of N points correspondent to the N original points in input space.
isomap advantages
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.
isomap disadvantages
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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