A Generic Unfolding Algorithm for Manifolds Estimated by Local Linear Approximations

The individual stages of most popular manifold learning algorithms are complicated by overlapping ideas – often consisting of a mix of learning how to embed, unfold and reduce the dimension of the manifold at the same time. Furthermore, the effect each step has on the final result is in many cases not clear. Research in both machine learning and mathematical communities has focused on the steps involved in manifold embedding and estimation, and sample sizes and performance bounds related to these operations have been explored. However, the problem of unwrapping or unfolding manifolds has received relatively little attention despite being an integral part of manifold learning in general. In this work, we present a new generic algorithm for unfolding manifolds that have been estimated by local linear approximations. Our algorithm is a combination of ideas from principal curves and density ridge estimation and tools from classical differential geometry. Numerical experiments on both real and synthetic data sets illustrates the merit of our proposed algorithm.