Contents I. Introduction
Over the past decade, numerous manifold learning methods have been proposed for nonlinear dimensionality reduction (NLDR). From methodology, these methods can be divided into two categories: global algorithms and local algorithms. Representative global algorithms contain isometric mapping [1], maximum variance unfolding [2] and local coordinates alignment with global preservation [3]. Local methods mainly include Laplacian eigenmaps (LEM)[4], locally linear embedding (LLE)[5], Hessian eigenmaps (HLLE)[6], local tangent space alignment [7], local linear transformation embedding [8], stable local approaches [9], and maximal linear embedding [10]. Different local approaches try to learn different geometric information of the underlying manifold, since they are developed based on the knowledge and experience of experts for their own purposes [11]. Thus, each existing local method only learns partial information of the true underlying manifold from which the datasets are sampled. Therefore, it is essential and more informative to provide a common framework for synthesizing the geometric information extracted from different local methods to better discover the underlying manifold structure. In this letter, we introduce a novel method to unify the local manifold learning algorithms (e.g. LEM, LLE and HLLE). Inspired by HLLE which utilizes local tangent coordinates to estimate the local Hessian, we propose to use local tangent coordinates to estimate the local objects defined in different local methods. Then, we employ the selection matrix to connect the local objects with a global functional. Finally, we develop an alternating optimization-based algorithm to discover the global coordinate system of lower dimensionality.
