Isomap

Isomap is highly efficient and generally applicable to a broad range of data sources and dimensionalities.

Isomap is one representative of isometric mapping methods, and extends metric multidimensional scaling (MDS) by incorporating the geodesic distances imposed by a weighted graph.

Isomap is distinguished by its use of the geodesic distance induced by a neighborhood graph embedded in the classical scaling.

Isomap defines the geodesic distance to be the sum of edge weights along the shortest path between two nodes (computed using Dijkstra's algorithm, for example).

The connectivity of each data point in the neighborhood graph is defined as its nearest k Euclidean neighbors in the high-dimensional space.

[4] Even a single short-circuit error can alter many entries in the geodesic distance matrix, which in turn can lead to a drastically different (and incorrect) low-dimensional embedding.

Isomap on the “Swiss roll” data set. (A) Two points on the Swiss roll and their geodesic curve. (B) The KNN graph (with K = 7 and N = 2000) allows a graph geodesic (red) that approximates the smooth geodesic. (C) The Swiss roll "unrolled", showing the graph geodesic (red) and the smooth geodesic (blue). Replication of Figure 3 of [ 1 ] .