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Title: Probabilistic principal curves on Riemannian manifolds Authors:  Seungwoo Kang - Seoul National University (Korea, South) [presenting]
Hee-Seok Oh - Seoul National University (Korea, South)
Abstract: A new curve fitting approach is studied that is useful for the representation and dimension reduction of data on Riemannian manifolds. We extend the probabilistic formulation of the curve passing through the middle of data on Euclidean space to Riemannian symmetric space. To this end, we define a principal curve based on a mixture model for observations and unobserved latent variables, and propose a new algorithm to estimate the principal curve for given data points on Riemannian manifolds using a series of procedures in `unrolling, unwrapping, and wrapping' and EM algorithm. Some properties for justification of the estimation algorithm are further investigated. Results from numerical examples, including several simulation sets on hyperbolic space, sphere, special orthogonal group, and a real data example, demonstrate the promising empirical properties of the proposed probabilistic approach.