Principal Boundary on Riemannian Manifolds

TL;DR

This paper introduces the principal boundary, a novel method for classification on Riemannian manifolds that finds an optimal curve between classes by maximizing margin, extending SVM concepts to nonlinear manifold data.

Contribution

The paper proposes the principal boundary, a new intrinsic classification boundary on Riemannian manifolds, with theoretical properties and practical applications demonstrated.

Findings

Principal boundary coincides locally with SVM decision boundary.

The method has proven optimality and convergence properties.

Application to real data illustrates its effectiveness.

Abstract

We consider 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 ambient space, we aim to acquire a classification boundary for the classes with labels, using the intrinsic metric on the manifolds. Motivated by finding an optimal boundary between the two classes, we invent a novel approach -- the principal boundary. From the perspective of classification, the principal boundary is defined as an optimal curve that moves in between the principal flows traced out from two classes of 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. Some optimality and convergence properties of the random principal boundary and its population counterpart are also shown. We illustrate how to find, use and interpret the principal boundary with an application in real data.