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Abstract:
I present a new computational-geometric framework for
unsupervised mapping of a manifold of perceptual observations.
Nonlinear dimensionality reduction is formulated here as the
problem of trying to find a Euclidean feature-space embedding of a
set of observations that preserves as closely as possible their
intrinsic metric structure, i.e. the distances between points on
the observation manifold as measured along geodesic paths. Our
isometric feature mapping procedure ("isomap") is able to reliably
recover low-dimensional nonlinear structure in realistic perceptual
data sets, such as a manifold of face images, where conventional
global mapping methods find only local minima. The recovered map
provides a canonical set of globally meaningful features, which
allows perceptual transformations such as interpolation,
extrapolation, and analogy -- highly nonlinear transformations in
the original observation space -- to be computed with simple linear
operations in feature space.
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