To generalize Principal Component Analysis (PCA) to Riemannian manifolds, we first propose barycentric subspaces, implicitly defined as the locus of weighted means of k+1 reference points. This locally defines submanifolds of dimension k which can naturally be nested, allowing the construction of inductive forward or backward nested subspaces minimizing the unexplained variance. To optimize the whole hierarchy consistently, Barycentric Subspaces Analysis (BSA) proposes to optimize the accumulated unexplained variance on flags of linear subspaces.

This presentation is part of Minisymposium “MS63 - Geometric methods for shape analysis with applications to biomedical imaging and computational anatomy”
organized by: Martin Bauer (Florida State University) , Nicolas Charon (Johns Hopkins University) .

Authors:

Xavier Pennec (Université Côte d'Azur and Inria)

Keywords:

geometric statistics, nonlinear optimization, statistics on manifolds