Isomap

Isomap (Isometric Mapping) is a nonlinear dimensionality reduction algorithm introduced by Tenenbaum, de Silva, and Langford in 2000. It extends classical multidimensional scaling by preserving not Euclidean distances but geodesic distances — the shortest paths along the manifold on which the data is assumed to lie.

The algorithm proceeds in three steps. First, it constructs a nearest neighbor graph connecting each point to its closest neighbors in the high-dimensional space. Second, it computes shortest paths between all pairs of points on this graph, approximating the true geodesic distances along the manifold. Third, it applies classical multidimensional scaling to these geodesic distances to find a low-dimensional embedding.

Isomap was a foundational contribution to manifold learning, demonstrating that nonlinear structure invisible to PCA could be recovered by respecting the intrinsic geometry of the data. It has since been superseded by faster methods like t-SNE and UMAP, but its conceptual framework — that data lives on a curved surface and distance must be measured along that surface, not through the ambient space — remains central to modern dimensionality reduction.