We present an approach to the data reduction problem which is based on a
well-known, constructive proof of Whitney's embedding theorem. This
approach involves picking projections of the high-dimensional system
which are optimized in the sense that they are easy to invert, i.e., the
inverse is designed to have a small Lipschitz constant. This is
accomplished by considering the effect of the projections on the set of
unit secants constructed from the data. We propose optimality criteria
for identifying good projections and algorithms to find them. We
illustrate this methodology via several applications including the
reduction of high-dimensional dynamical systems, signal and image
processing. This is joint work with David Broomhead, Department of
Mathematics, UMIST, UK.
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