Sungkyu Jung
Department Statistics and Operations Research
University of North Carolina at Chapel Hill
Statistical Analysis of Data on Curved Manifolds
A number of interesting data lie naturally on curved manifolds, where
conventional statistics are sometimes not directly applicable. This type of
data arises in, for example, shape and image analysis. In this talk, I will
discuss some challenges in statistical analysis on these non-Euclidean
feature spaces, and introduce some data analytic methods that generalize
principal component analysis (PCA). We first focus on high dimensional
spheres that are highly important in many applications, and introduce a
general framework for a novel non-geodesic (non-linear) decomposition. This
decomposition finds a sequence of sub-manifolds with decreasing dimensions,
which can be interpreted as an analogue of PCA.
The method is adapted and extended to more complex manifolds: shape spaces
and direct product manifold (Cartesian product of simpler manifolds). The
method provides a coordinate system to visualize the data structure, and an
intuitive summary of principal modes of variation, as exemplified by several
interesting real data sets.
In addition, I will continue to discuss some asymptotic results on
Euclidean PCA, to illustrate conditions under which PCA is informative in
high dimension, low sample size situations.
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