Atlas Compatibility Transformation: A Normal Manifold Learning Algorithm

Over the past few years, nonlinear manifold learning has been widely exploited in data analysis and machine learning. This paper presents a novel manifold learning algorithm, named atlas compatibility transformation (ACT). It solves two problems which correspond to two key points in the manifold definition: how to chart a given manifold and how to align the patches to a global coordinate space based on compatibility. For the first problem, we divide the manifold into maximal linear patch (MLP) based on normal vector field of the manifold. For the second problem, we align patches into an optimal global system by solving a generalized eigenvalue problem. Compared with the traditional method, the ACT could deal with noise datasets and fragment datasets. Moreover, the mappings between high dimensional space and low dimensional space are given. Experiments on both synthetic data and real-world data indicate the effection of the proposed algorithm.