Title: Principal Boundary on Riemannian Manifolds,
Abstract: We revisit the classification problem and focus on nonlinear methods for
classification on manifolds. For multivariate datasets lying on an embedded
nonlinear Riemannian manifold within the higher-dimensional space, our aim is
to acquire a classification boundary between the classes with labels. Motivated
by the principal flow [Panaretos, Pham and Yao, 2014], a curve that moves along
a path of the maximum variation of the data, we introduce the principal
boundary. From the classification perspective, the principal boundary is
defined as an optimal curve that moves in between the principal flows traced
out from two classes of the data, and at any point on the boundary, it
maximizes the margin between the two classes. We estimate the boundary in
quality with its direction supervised by the two principal flows. We show that
the principal boundary yields the usual decision boundary found by the support
vector machine, in the sense that locally, the two boundaries coincide. By
means of examples, we illustrate how to find, use and interpret the principal
boundary.