Related papers: Principal subbundles for dimension reduction

Principal subbundles for dimension reduction

URL: http://arxiv.org/abs/2307.03128v1
Date: Thu, 6 Jul 2023 16:55:21 GMT
Title: Principal subbundles for dimension reduction
Authors: Morten Akh{\o}j, James Benn, Erlend Grong, Stefan Sommer, Xavier Pennec
Abstract summary: We show how sub-Riemannian geometry can be used for manifold learning and surface reconstruction. We show that the framework is robust when applied to noisy data.
Score: 0.07515511160657122
License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
Abstract: In this paper we demonstrate how sub-Riemannian geometry can be used for manifold learning and surface reconstruction by combining local linear approximations of a point cloud to obtain lower dimensional bundles. Local approximations obtained by local PCAs are collected into a rank $k$ tangent subbundle on $\mathbb{R}^d$, $k<d$, which we call a principal subbundle. This determines a sub-Riemannian metric on $\mathbb{R}^d$. We show that sub-Riemannian geodesics with respect to this metric can successfully be applied to a number of important problems, such as: explicit construction of an approximating submanifold $M$, construction of a representation of the point-cloud in $\mathbb{R}^k$, and computation of distances between observations, taking the learned geometry into account. The reconstruction is guaranteed to equal the true submanifold in the limit case where tangent spaces are estimated exactly. Via simulations, we show that the framework is robust when applied to noisy data. Furthermore, the framework generalizes to observations on an a priori known Riemannian manifold.

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