Isomap "nonlinear dimensionality reduction" numbre of points

Question

I have a question please , it's about 'Isomap' nonlinear dimensionality reduction, in normal cases when I introduce a matrix distance of 100 * 100 and I apply Isomap [http://isomap.stanford.edu/][1] I get the coordinates of 100 points ,in other cases I do not understand why, with a matrix of 150 * 150 i obtain juste 35 or 50 points ?

Answer 1

The first step of Isomap is usually to create a "nearest neighbor matrix" so that every point is connected to its 4 or 6 or 8 or something nearest neighbors.

So, you may start with a distance matrix that is 100 x 100 and every point has a distance to 99 other points, after this first step the distances for anything but the (4 or 6 or 8) closest points are set to infinity.

Then Isomap computes a shortest path distance, hopping between nearby points to get to farther away points.

In your case, when you create a matrix of 150 points, I think that once you only keep the nearby points in the first step, the points become disconnected, and there is path between distant points. The default behavior of many Isomap codes is to return the Isomap embedding of the largest collection of connected points.

How can you fix this? 1. You can increase the number of nearest neighbors that you use until you get all the points included.

Caveat: In many natural cases, if you include most or all neighbors, this ends up in the case where the shortest-path part of the procedure does nothing, and this reduces to a problem called "multi-dimensional scaling" which gives a linear embedding.