Local non-linear alignment for non-linear dimensionality reduction

In manifold learning, alignment is performed with the objective of deriving the global low-dimensional coordinates of input data from their local coordinates. In virtually all alignment processes, the relation between the local and global coordinates is designed intuitively, without mathematical deduction and detailed analysis. In this study, the authors propose a local non-linear alignment manifold learning algorithm (LNA) for non-linear dimensionality reduction, based on the concept of local pullback and the mathematical characteristics of a manifold. According to mathematical manifold theory, a function defined on a manifold cannot be differentiated directly on the manifold directly. Instead, it has to be pulled back to Euclidean space with the help of local homeomorphism between the manifold and Euclidean space, where it is then differentiated. In the authors' proposed algorithm, the component functions of global homeomorphism are regarded as the functions defined on the manifold and pulled back to the Euclidean space. Then, Taylor expansion is utilised up to the second order to establish the relation between the global and local coordinates. The objective function in LNA is based on the alignment error and can be solved with an eigenvalue problem. The experimental results conducted on various datasets verify the validity of the authors' method.