  In the recent past, nested structures in Riemannian manifolds has been
studied in the context of dimensionality reduction as an alternative to the
popular principal geodesic analysis (PGA) technique, for example, the principal
nested spheres. In this paper, we propose a novel framework for constructing a
nested sequence of homogeneous Riemannian manifolds. Common examples of
homogeneous Riemannian manifolds include the $n$-sphere, the Stiefel manifold,
the Grassmann manifold and many others. In particular, we focus on applying the
proposed framework to the Grassmann manifold, giving rise to the nested
Grassmannians (NG). An important application in which Grassmann manifolds are
encountered is planar shape analysis. Specifically, each planar (2D) shape can
be represented as a point in the complex projective space which is a complex
Grass-mann manifold. Some salient features of our framework are: (i) it
explicitly exploits the geometry of the homogeneous Riemannian manifolds and
(ii) the nested lower-dimensional submanifolds need not be geodesic. With the
proposed NG structure, we develop algorithms for the supervised and
unsupervised dimensionality reduction problems respectively. The proposed
algorithms are compared with PGA via simulation studies and real data
experiments and are shown to achieve a higher ratio of expressed variance
compared to PGA.
