Enhanced Multilevel Manifold Learning

Two multilevel frameworks for manifold learning algorithms are discussed which are based on an affinity graph whose goal is to sketch the neighborhood of each sample point. One framework is geometric and is suitable for methods aiming to find an isometric or a conformal mapping, such as isometric feature mapping (Isomap) and semidefinite embedding (SDE). The other is algebraic and can be incorporated into methods that preserve the closeness of neighboring points, such as Locally Linear Embedding (LLE) and Laplacian eigenmaps (LE). The multilevel coarsening technique presented in this paper can be applied to both directed and undirected graphs. It is based on the degree of dependency between vertices, a parameter that is introduced to control the speed of coarsening. In the algebraic framework, we can coarsen the problem by some form of restriction and then uncoarsen the solution by prolongation, as a standard procedure in multigrid methods. In the geometric framework, the uncoarsening method can improve the embedding quality in the sense of being isometric or conformal. The methods presented provide multiscale resolution of a manifold, and a coarse embedding is very inexpensive. An application to intrinsic dimension estimation is illustrated. Results of experiments on synthetic data and two image data sets are presented.