Master isometric mapping for preserving global geodesic distances. Learn how Isomap constructs neighbor graphs, computes shortest paths, and applies MDS to preserve manifold topology.
Isomap (Isometric Mapping) is a nonlinear dimensionality reduction method that preserves geodesic distances (shortest paths along the manifold) rather than Euclidean distances. It addresses the limitation of linear methods that compute "straight-line" distances in high-dimensional space, which don't reflect the true structure of curved manifolds.
On a curved manifold, the Euclidean distance (straight line) between two points may be misleading. The geodesic distance (shortest path along the manifold surface) better reflects true similarity.
Example: On Earth's surface, the geodesic distance between two cities is the great circle route, not a straight line through the Earth.
The geodesic distance between two points on a manifold is the length of the shortest path that stays on the manifold surface.
Isomap approximates geodesic distances by:
The complete Isomap algorithm:
For each sample , find its k nearest neighbors based on Euclidean distance. Connect to its neighbors with edges weighted by Euclidean distance. Non-neighbors have infinite distance.
Use shortest path algorithm (Dijkstra or Floyd-Warshall) to compute shortest path distances between all pairs of samples in the neighbor graph. These approximate geodesic distances .
Use the geodesic distance matrix as input to MDS algorithm. MDS finds low-dimensional embeddings such that.