Advances in Geometric Statistics for Manifold Dimension Reduction

Geometric statistics aim at shifting the classical paradigm for inference from points in a Euclidean space to objects living in a non-linear space, in a consistent way with the underlying geometric structure considered. In this chapter, we illustrate some recent advances of geometric statistics for dimension reduction in manifolds. Beyond the mean value (the best zero-dimensional summary statistics of our data), we want to estimate higher dimensional approximation spaces fitting our data. We first define a family of natural parametric geometric subspaces in manifolds that generalize the now classical geodesic subspaces: barycentric subspaces are implicitly defined as the locus of weighted means of k + 1 reference points with positive or negative weights summing up to one. Depending on the definition of the mean, we obtain the Fréchet, Karcher or Exponential Barycentric subspaces (FBS/KBS/EBS). The completion of the EBS, called the affine span of the points in a manifold is the most interesting notion as it defines complete sub-(pseudo)-spheres in constant curvature spaces. Barycentric subspaces can be characterized very similarly to the Euclidean case by the singular value decomposition of a certain matrix or by the diagonalization of the covariance and the Gram matrices. This shows that they are stratified spaces that are locally manifolds of dimension k at regular points. Barycentric subspaces can naturally be nested by defining an ordered series of reference points in the manifold. This allows the construction of inductive forward or backward properly nested sequences of subspaces approximating data points. These flags of barycentric subspaces generalize the sequence of nested linear subspaces (flags) appearing in the classical Principal Component Analysis. We propose a criterion on the space of flags, the accumulated unexplained variance (AUV), whose optimization exactly lead to the PCA decomposition in Euclidean spaces. This procedure is called barycentric subspace analysis (BSA). We illustrate the power of barycentric subspaces in the context of cardiac imaging with the estimation, analysis and reconstruction of cardiac motion from sequences of images.