Principal Manifolds and Nonlinear Dimension Reduction Via Local Tangent Space Alignment is a article published in Society for Industrial and Applied Mathematics on July 25th, 2006. The focus for this society is applied, computational and industrial mathematics. It is composed of people from a wide variety of vocations. Its members include engineersscientists, industrial mathematicians, and academic mathematicians. The society is active in promoting the use of analysis and modeling in all settings. The society also strives to support and provide guidance to educational institutions wishing to promote applied mathematics[1]. This article is written by Zhenyue Zhang and Hongyuan Zha. Professor Zhenyue Zhang is from department of Mathematics, Zhejiang University, while professor Hongyuan Zha comes from department of Computer Science and Engineering in Pennsylvania State University.