Out-of-sample generalizations for supervised manifold learning for classification

Supervised manifold learning methods for data classification map data samples residing in a high-dimensional ambient space to a lower-dimensional domain in a structure-preserving way, while enhancing the separation between different classes in the learned embedding. Most nonlinear supervised manifold learning methods compute the embedding of the manifolds only at the initially available training points, while the generalization of the embedding to novel points, known as the out-of-sample extension problem in manifold learning, becomes especially important in classification applications. In this work, 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 (RBF) 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 a progressive procedure. 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

A perturbation based out-of-sample extension framework

Out-of-sample extension is an important task in various kernel based non-linear dimensionality reduction algorithms. In this paper, we derive a perturbation based extension framework by extending results from classical perturbation theory. We prove that our extension framework generalizes the well-known Nystr{\"o}m method as well as some of its variants. We provide an error analysis for our extension framework, and suggest new forms of extension under this framework that take advantage of the structure of the kernel matrix. We support our theoretical results numerically and demonstrate the advantages of our extension framework both on synthetic and real data.Comment: 22 pages, 9 figure

A Generalised Solution to the Out-of-Sample Extension Problem in Manifold Learning

Manifold learning is a powerful tool for reducing the dimensionality of a dataset by finding a low-dimensional embedding that retains important geometric and topological features. In many applications it is desirable to add new samples to a previously learnt embedding, this process of adding new samples is known as the out-of-sample extension problem. Since many manifold learning algorithms do not naturally allow for new samples to be added we present an easy to implement generalized solution to the problem that can be used with any existing manifold learning algorithm. Our algorithm is based on simple geometric intuition about the local structure of a manifold and our results show that it can be effectively used to add new samples to a previously learnt embedding. We test our algorithm on both artificial and real world image data and show that our method significantly out performs existing out-of-sample extension strategies