Some of the widely used method are based on a similarity graph made with the local structure. For instance LLE uses the relative distances, which is related to similarities. Using similarities allows the use of sparse techniques. Indeed, a lot of points are not similar, and then the similarities matrix is sparse. This also means that a lot of manifold can be reduced with these techniques, but not with Isomap or the other geodesic-based techniques.
It is worth mentioning that I only implemented Laplacian Eigenmaps with a sparse matrix, due to the lack of generalized eigensolver for sparse matrix, but it will be available in a short time, I hope.