1-Introduction

       This article propose a new algorithm (LTSA) for nonlinear manifold learning and nonlinear dimension reduction. it’s tite is “PRINCIPAL MANIFOLDS AND NONLINEAR DIMENSION REDUCTION VIA LOCAL TANGENT SPACE ALIGNMENT ”. Was Published in 2002 by SIAM Journal of Scientific . The SIAM Journal on Scientific Computing  contains research articles on numerical methods and techniques for scientific computation. Papers address computational issues relevant to the solution of scientific or engineering problems and include computational results demonstrating the effectiveness of the proposed techniques. It was written by Zhenyue Zhang , et Hongyuan Zha .
hongyuan Zha was a faculty member of the Department of Computer Science and Engineering at Pennsylvania State University from 1992 to 2006, and he worked from 1999 to 2001 at Inktomi Corporation. Zha’s current research interests include computational mathematics and machine learning applications.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      Zhenyue Zhang is a mathematical researcher in the Department of Mathematics Research at Zhejiang • China University.    The authors had presented, in this article manifold learning and dimension reduction and he discusses the issue of learning local geometry using tangent spaces, then he shows how to align those local tangent spaces in order to learn the global coordinate system of the underlying manifold, finaly he based for algorithm local tangent space alignment (LTSA) algorithm and comparite with the local linear embedding (LLE)in order to resume the article, I start with exposing the scientific context at first, secondly, I will try to highlight the most important existing works; and the position of the article. Thirdly, the contribution in the scientific world, and finally I end my resume with showing the results and experiments of the method

2 - Context of the work

     In recent years, a variety of nonlinear dimensionality reduction techniques have been proposed that aim to overcome the limitations of traditional techniques such as PCA  and other dimension reduction .
The manifold learning  is the approach how attempts to reduce dimension of the dataset to ease representation and interpretation (we called the approach representation learning) The main issue with the high dimensional dataset is more difficult it becomes to sample the space. This causes many problems. Algorithms that operate on high-dimensional data tend to have a very high time complexity. Many machine learning algorithms, for example, struggle with high-dimensional data. In this article the authors speak about principale manifolds and nonlinear dimention reduction via local tangent space alignmentso what’s what’s local tangent aligment ??and why we use it?
            LTSA is technique that describes local properties of the high-dimensional data using the local tangent space of each datapoint . LTSA is based on the observation that, if local linearity of the manifold is assumed, there exists a linear mapping from a high-dimensional datapoint to its local tangent space, and that there exists a linear mapping from the corresponding low-dimensional data point to the same local tangent space . LTSA attempts to align these linear mappings in such a way, that they construct the local tangent space of the manifold from the low-dimensional representation. In other words, LTSA simultaneously searches for the coordinates of the low-dimensional data representations, and for the linear mappings of the low dimensional datapoints to the local tangent space of the high-dimensional data.\ref{391140}
        all manifold learning algorithms can be applied to data dimensionality reduction, producing a low-dimensional encoding from a high-dimensional one  and  among these algorithm LTSA .His principe is to to use the tangent space in the neighborhood of a data point to represent the local geometry, and then align those local tangent spaces to construct the global coordinate system for the nonlinear manifold.