Algorithm for Riemannian manifold learning

Abstract

We present a novel method for manifold learning, which identifies the low-dimensional manifold-like structure presented in a set of data points in a possibly high-dimensional space with homomorphism contained. The main idea is derived from the concept of covariant components in curvilinear coordinate systems. In a linearly transparent way, we translate this idea to a cloud of data points in order to calculate the coordinates of the points directly. Our implementation currently uses Dijkstra's algorithm for shortest paths in graphs and some basic theorems from Riemannian differential geometry. We expect this approach to open up new possibilities for manifold learning using only geometry constraints, which means the coordinate system is "learned" from experimental high-dimensional data rather than defined analytically using e.g. models based on PCA, MDS, and Eigenmaps.