Out-of-Sample Generalizations for Supervised Manifold Learning for Classification

Supervised manifold learning methods for data classification map high-dimensional data samples to a lower dimensional domain in a structure-preserving way while increasing the separation between different classes. Most manifold learning methods compute the embedding only of the initially available data; however, the generalization of the embedding to novel points, i.e., the out-of-sample extension problem, becomes especially important in classification applications. In this paper, we propose a semi-supervised method for building an interpolation function that provides an out-of-sample extension for general supervised manifold learning algorithms studied in the context of classification. The proposed algorithm computes a radial basis function interpolator that minimizes an objective function consisting of the total embedding error of unlabeled test samples, defined as their distance to the embeddings of the manifolds of their own class, as well as a regularization term that controls the smoothness of the interpolation function in a direction-dependent way. The class labels of test data and the interpolation function parameters are estimated jointly with an iterative process. Experimental results on face and object images demonstrate the potential of the proposed out-of-sample extension algorithm for the classification of manifold-modeled data sets.