The paper presents a novel algorithm for manifold learning and nonlinear dimensionality reduction, focusing on aligning tangent spaces to reconstruct the global coordinates of data points on a parameterized manifold. The authors, ZHANG Zhen-yue and ZHA Hong-yuan, introduce the Local Tangent Space Alignment (LTSA) algorithm, which constructs local geometric structures using tangent spaces at each data point and aligns these spaces to form a global coordinate system. The method is particularly useful for high-dimensional data sets that lie close to a low-dimensional nonlinear manifold, such as image vectors and document vectors. The paper includes an error analysis, demonstrating that reconstruction errors can be small under certain conditions, and discusses theoretical and algorithmic issues for further research. The authors emphasize the interplay between nonlinear dimensionality reduction and manifold reconstruction, highlighting the importance of both processes in understanding the underlying structure of the data.The paper presents a novel algorithm for manifold learning and nonlinear dimensionality reduction, focusing on aligning tangent spaces to reconstruct the global coordinates of data points on a parameterized manifold. The authors, ZHANG Zhen-yue and ZHA Hong-yuan, introduce the Local Tangent Space Alignment (LTSA) algorithm, which constructs local geometric structures using tangent spaces at each data point and aligns these spaces to form a global coordinate system. The method is particularly useful for high-dimensional data sets that lie close to a low-dimensional nonlinear manifold, such as image vectors and document vectors. The paper includes an error analysis, demonstrating that reconstruction errors can be small under certain conditions, and discusses theoretical and algorithmic issues for further research. The authors emphasize the interplay between nonlinear dimensionality reduction and manifold reconstruction, highlighting the importance of both processes in understanding the underlying structure of the data.