In page 8 of the paper Laplacian Eigenmaps for dimensionality reduction and Data Representation it reads:
Standard methods show that the solution is provided by the matrix of eigenvectors corresponding to the lowest eigenvalues of the generalized eigenvalue problem...
Can anyone explain why in this representation the "lowest eigenvalues" correspond to eigen-vectors which capture more information on the manifold? This is counter-intuitive to what happens in standard eigen-methods such as PCA.
I would also love an intuitive explanation to the isoperimetric/geometric properties of the high eigenvectors/eigenvalues, since these appear to be easier to compute for the problem I am facing (A large, sparse graph).
Any help is appreciated, thanks!