Nested Grassmannians for Dimensionality Reduction with Applications

TL;DR

This paper introduces nested Grassmannians, a new framework for dimensionality reduction on Riemannian manifolds, particularly applied to shape analysis, outperforming principal geodesic analysis in variance expression.

Contribution

It proposes a novel nested manifold framework that explicitly leverages geometry, enabling effective dimensionality reduction without requiring geodesic submanifolds.

Findings

Higher variance explained compared to PGA

Effective in shape analysis applications

Applicable to various homogeneous Riemannian manifolds

Abstract

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.