In this article, first it introduces local tangent space alignment(LTSA) algorithm to modeled high-dimensional data by using low-dimensional nonlinear manifold in section 1. Second it formulates the problem of manifold learning and dimension reduction, and illustrates the intricacy of the problem using the linear case in section 2. In section 3 it discusses the issue of learning local geometry by using tangent space. Next in section 4, it shows how to align those local tangent spaces in order to learn the global coordinate system of the underlying manifold. It discusses how to construct the manifold once the global coordinate system is available in section 5. Then it presents an error analysis of LTSA in section 6. In section 7, it shows how the partial eigen-decomposition used in global coordinate construction could be efficiently computed. Then it presents a collection of numerical experiments in section 8. Finally in section 9, it concludes the paper and addresses several theoretical and algorithmic issues for further research and improvements[2]