We outline a general framework for a geometric approach to data analysis.
The basic setting is one where one assumes that data is generated from
a probability distribution that is concentrated on or near a submanifold of
Euclidean space. A variety of problems such as dimensionality reduction,
regression, pattern classification and so on may then be formulated in
this setting and reduce to estimating various functional maps on this
(unknown) submanifold. Additionally we will also discuss algorithms for
estimating other geometric and topological invariants of the submanifold
from sampled data (such as dimension, connected components and
more generally the homology).
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