Part 2 of 2 on recasting principal components.
Here the Gram matrix is related to the matrix of squared distances (sometimes called the Euclidean Distance matrix).
This means if we begin with distances (however arrived at) we can then use principal components (or metric scaling) to find a dimension reduction.
Use distances, for example, as defined by following he shortest path between points on some nearest neighbour graph. This should (for suitable choice of neighbourhood) be much like following whatever nonlinear manifold the data may be near.
Use these distances to determine the Gram matrix and then do dimension reduction.
A particular instance of this approach is called ISOMAP.
Frey faces are again used as an example with ISOMAP and interactive graphics using loon, navigation graphs, and scagnostics.
