Graduate Seminar: Pavlos Zoubouloglou
Dimension Reduction on Manifolds with an Emphasis on the Torus
Recent advancements in the way information is collected have brought along the need to analyze data of a non-Euclidean nature. Dimension reduction has proved to be a challenging task for data that live on high dimensional directional manifolds (e.g. spheres and torii), as PCA disregards their inherent periodicity. For the first part of this talk, an overview is provided of methods that have been developed to perform dimension reduction for data that lie on general directional manifolds, including tangent plane-based methods and geodesic-based methods. In the context of spherical data, Principal Nested Spheres provides a complete analogue of PCA and demonstrates that manifold-specific methods can outperform generic methods. For the second part of the talk, we review methods that pertain to dimension reduction on torii. A novel approach to perform dimension reduction for toroidal data, referred to as Scaled Torus Principal Component Analysis (STPCA), is introduced. Two data applications in molecular biology and astronomy show that STPCA outperforms existing methods for the investigated datasets.