Rehabilitating Isomap: Euclidean Representation of Geodesic Structure
Abstract
Manifold learning techniques for nonlinear dimension reduction assume that highdimensional feature vectors lie on a lowdimensional manifold, then attempt to exploit manifold structure to obtain useful lowdimensional Euclidean representations of the data. Isomap, a seminal manifold learning technique, is an elegant synthesis of two simple ideas: the approximation of Riemannian distances with shortest path distances on a graph that localizes manifold structure, and the approximation of shortest path distances with Euclidean distances by multidimensional scaling. We revisit the rationale for Isomap, clarifying what Isomap does and what it does not. In particular, we explore the widespread perception that Isomap should only be used when the manifold is parametrized by a convex region of Euclidean space. We argue that this perception is based on an extremely narrow interpretation of manifold learning as parametrization recovery, and we submit that Isomap is better understood as constructing Euclidean representations of geodesic structure. We reconsider a wellknown example that was previously interpreted as evidence of Isomap's limitations, and we reexamine the original analysis of Isomap's convergence properties, concluding that convexity is not required for shortest path distances to converge to Riemannian distances.
 Publication:

arXiv eprints
 Pub Date:
 June 2020
 DOI:
 10.48550/arXiv.2006.10858
 arXiv:
 arXiv:2006.10858
 Bibcode:
 2020arXiv200610858T
 Keywords:

 Statistics  Machine Learning;
 Computer Science  Machine Learning;
 62H99
 EPrint:
 27 pages, 4 figures