# Principal Boundary on Riemannian Manifolds

arXiv: https://arxiv.org/abs/1711.06705 · HTML: https://mlsanity.com/paper/1711.06705/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 21 Oct 2017.

## Built on by (most cited first, 3 of 3)

- [Manifold learning and optimization using tangent space proxies](https://mlsanity.com/paper/2501.12678/) (2025)
- [Consistent Interpolating Ensembles via the Manifold-Hilbert Kernel](https://mlsanity.com/paper/2205.09342/) (2022)
- [Atlas-based Manifold Representations for Interpretable Riemannian Machine Learning](https://mlsanity.com/paper/2510.17772/) (2025)

## Abstract (arXiv)

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.