# Rehabilitating Isomap: Euclidean Representation of Geodesic Structure

arXiv: https://arxiv.org/abs/2006.10858 · HTML: https://mlsanity.com/paper/2006.10858/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 18 Jun 2020.

## Built on by (most cited first, 5 of 5)

- [A singular Riemannian geometry approach to Deep Neural Networks I. Theoretical foundations](https://mlsanity.com/paper/2201.09656/) (2021)
- [Continuous Multidimensional Scaling](https://mlsanity.com/paper/2402.04436/) (2024)
- [Finding Significant Features for Few-Shot Learning using Dimensionality Reduction](https://mlsanity.com/paper/2107.06992/) (2021)
- [Semisupervised regression in latent structure networks on unknown manifolds](https://mlsanity.com/paper/2305.02473/) (2023)
- [Learning Submanifolds for Subsequent Inference on Random Dot Product Graphs, Part 1: Theory](https://mlsanity.com/paper/2609.19357/) (2026)

## Abstract (arXiv)

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.