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Principal Boundary on Riemannian Manifolds
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.Comment: 31 pages,10 figure
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