Representing images and videos with Symmetric Positive Definite (SPD) matrices and considering the Riemannian geometry of the resulting space has proven beneficial for many recognition tasks. Unfortunately, computation on the Riemannian manifold of SPD matrices --especially of high-dimensional ones-- comes at a high cost that limits the applicability of existing techniques. In this paper we introduce an approach that lets us handle high-dimensional SPD matrices by constructing a lower-dimensional, more discriminative SPD manifold. To this end, we model the mapping from the high-dimensional SPD manifold to the low-dimensional one with an orthonormal projection. In particular, we search for a projection that yields a low-dimensional manifold with maximum discriminative power encoded via an affinity-weighted similarity measure based on metrics on the manifold. Learning can then be expressed as an optimization problem on a Grassmann manifold. Our evaluation on several classification tasks shows that our approach leads to a significant accuracy gain over state-of-the-art methods.
A general framework for a novel non-geodesic decomposition of high-dimensional spheres or high-dimensional shape spaces for planar landmarks is discussed. The decomposition, principal nested spheres, leads to a sequence of submanifolds with decreasing intrinsic dimensions, which can be interpreted as an analogue of principal component analysis. In a number of real datasets, an apparent one-dimensional mode of variation curving through more than one geodesic component is captured in the one-dimensional component of principal nested spheres. While analysis of principal nested spheres provides an intuitive and flexible decomposition of the high-dimensional sphere, an interesting special case of the analysis results in finding principal geodesics, similar to those from previous approaches to manifold principal component analysis. An adaptation of our method to Kendall's shape space is discussed, and a computational algorithm for fitting principal nested spheres is proposed. The result provides a coordinate system to visualize the data structure and an intuitive summary of principal modes of variation, as exemplified by several datasets. Copyright 2012, Oxford University Press.