Spectral methods for semi-supervised manifold learning

Given a finite number of data points sampled from a low-dimensional manifold embedded in a high dimensional space together with the parameter vectors for a subset of the data points, we need to determine the parameter vectors for the rest of the data points. This problem is known as semi-supervised manifold learning, and in this paper we propose methods to handle this problem by solving certain eigenvalue-problems. Our proposed methods address two key issues in semi-supervised manifold learning: 1) fitting of the local affine geometric structures, and 2) preserving the global manifold structures embodied in the overlapping neighborhoods around each data points. We augment the alignment matrix of local tangent space alignment (LTSA) with the orthogonal projection based on the known parameter vectors, giving rise to the eigenvalue problem that characterizes the semi-supervised manifold learning problem. We also discuss the roles of different types of neighborhoods and their influence on the learning process. We illustrate the performance of the proposed methods using both synthetic data sets as well as data sets arising from applications in video annotations.