Principal subbundles for dimension reduction
Morten Akhøj James Benn Erlend Grong Stefan Sommer Xavier Pennec
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.