3- Positioning :

The purpose of    LOCAL TANGENT SPACE ALIGNMENT  and other dimensionality reduction algorithms like Locally Linear Embedding (LLE) ), Principal Component Analysis (PCA) and Isomap, is to find correlations and connections between different samples of an observed object and plotting them into a drawn that can be visualized. Also it is able to maps the resource data into a single global coordinate system, optimizing it not using local minima and learning form nonlinear manifolds as it is shown in the source article    .
     Several different algorithms have been developed to dimensionality reduction so he  Produce a compact low-dimensional encoding from high-dimensiona , While all of these methods have a similar goal, approaches to the problem are different.
      we have linear and nolinear dimensionality we have a several methods use to dimentionality reduction for exemple :
a-Principal components analysis (PCA)
       Is a very popular technique for dimensionality reduction. PCA aims to find a linear subspace of dimension ā€œdā€ lower than ā€œnā€ such that the data points lie mainly on this linear subspace.