1. Introduction

Nonlinear dimensionality reduction (NLDR) plays an important role in the modern data analysis system, since many objects in our world can only be electronically represented with high-dimensional data such as images, videos, speech signals, and text documents. We usually need to analyze a large amount of data and process them, and however, it is very complicated or even infeasible to process these high-dimensional data directly, due to their high computational complexity on both time and space. Over the past decade, numerous manifold learning methods have been proposed for nonlinear dimensionality reduction. 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

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Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

[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 (LTSA) [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]. Therefore, only partial information from the true underlying manifold is learned by each existing local manifold learning method. Thus, to better discover the underlying manifold structure, it is more informative and essential to provide a common framework for synthesizing the geometric information extracted from different local methods. In this chapter, we propose an interesting method to unify the local manifold learning algorithms (e. g., LEM, LLE, HLLE, and LTSA). Inspired by HLLE which employs local tangent coordinates to compute the local Hessian, we propose to utilize local tangent coordinates to estimate the local objects defined in different local methods. Then, we employ the truncation function from differential manifold 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.
