Pre-image Problem in Manifold Learning and Dimensional Reduction Methods

Manifold learning and dimensional reduction methods provide a low dimensional embedding for a collection of training samples. These methods are based on the eigenvalue decomposition of the kernel matrix formed using the training samples. In [2] the embedding is extended to new test samples using the Nystrom approximation method. This paper addresses the pre-image problem for these methods, which is to find the mapping back from the embedding space to the input space for new test points. The relationship of these learning methods to kernel principal component analysis [6] and the connection of the out-of-sample problem to the pre-image problem [1] is used to provide the pre-image.