Spectral approximations in machine learning

In many areas of machine learning, it becomes necessary to find the eigenvector decompositions of large matrices. We discuss two methods for reducing the computational burden of spectral decompositions: the more venerable Nystom extension and a newly introduced algorithm based on random projections. Previous work has centered on the ability to reconstruct the original matrix. We argue that a more interesting and relevant comparison is their relative performance in clustering and classification tasks using the approximate eigenvectors as features. We demonstrate that performance is task specific and depends on the rank of the approximation.