The problem of nonlinear dimensionality reduction arises often in
statistics, machine learning, and pattern recognition. In this talk, I
will begin by reviewing two very different solutions to this problem,
one based on the symmetries of locally linear reconstructions, the
other based on semidefinite programming. The resulting algorithms can
be used to analyze high dimensional data that lies on or near a low
dimensional manifold. I will then illustrate how these algorithms work
on examples of curves and surfaces, as well as images of faces,
handwritten digits, and solid objects. Finally, I will describe recent
work in which the symmetries of the first algorithm are used to speed
up the second, often by several orders of magnitude.
Joint work with Kilian Weinberger and Ben Packer.
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