Nonlinear dimensionality reduction frameworks
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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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Outline l.jpg
Outline

  • Intuition of Nonlinear Dimensionality Reduction(NLDR)

  • ISOMAP

  • LLE

  • NLDR in Gait Analysis




Brain representation5 l.jpg
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 l.jpg
Manifold Learning

  • A manifold is a topological space which is locally Euclidean

  • An example of nonlinear manifold:


Manifold learning10 l.jpg

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 l.jpg
Linear Approach- PCA for data in high dimension.

  • PCA Finds subspace linear projections of input data.


Linear approach mds l.jpg
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.


Nonlinear approaches isomap l.jpg

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


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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 l.jpg
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 l.jpg
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 d l.jpg
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 l.jpg
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 l.jpg
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 l.jpg
PCA, MD vs ISOMAP for data in high dimension.


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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 l.jpg
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.