Riemannian Principal Component Analysis
This work addresses the challenge of analyzing datasets with region-specific or part-specific distance notions, such as in medical imaging or other structured data, by providing a unified approach that respects intrinsic geometric properties, though it builds incrementally on existing methods like Principal Geodesic Analysis.
The paper tackles the problem of extending Principal Component Analysis (PCA) to data on Riemannian manifolds, where traditional Euclidean assumptions fail, by introducing a generalized framework called Riemannian Principal Component Analysis (R-PCA) that adapts PCA methodology to incorporate local metrics and manifold geometry, enabling dimensionality reduction and statistical analysis directly on manifolds.
This paper proposes an innovative extension of Principal Component Analysis (PCA) that transcends the traditional assumption of data lying in Euclidean space, enabling its application to data on Riemannian manifolds. The primary challenge addressed is the lack of vector space operations on such manifolds. Fletcher et al., in their work {\em Principal Geodesic Analysis for the Study of Nonlinear Statistics of Shape}, proposed Principal Geodesic Analysis (PGA) as a geometric approach to analyze data on Riemannian manifolds, particularly effective for structured datasets like medical images, where the manifold's intrinsic structure is apparent. However, PGA's applicability is limited when dealing with general datasets that lack an implicit local distance notion. In this work, we introduce a generalized framework, termed {\em Riemannian Principal Component Analysis (R-PCA)}, to extend PGA for any data endowed with a local distance structure. Specifically, we adapt the PCA methodology to Riemannian manifolds by equipping data tables with local metrics, enabling the incorporation of manifold geometry. This framework provides a unified approach for dimensionality reduction and statistical analysis directly on manifolds, opening new possibilities for datasets with region-specific or part-specific distance notions, ensuring respect for their intrinsic geometric properties.