  Manifold learning techniques for nonlinear dimension reduction assume that
high-dimensional feature vectors lie on a low-dimensional manifold, then
attempt to exploit manifold structure to obtain useful low-dimensional
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 well-known example that was previously interpreted as evidence of
Isomap's limitations, and we re-examine the original analysis of Isomap's
convergence properties, concluding that convexity is not required for shortest
path distances to converge to Riemannian distances.
