The local tangent space provides a low-dimensional linear approximation of the local geometric structure of the nonlinear manifold. Those local tangent coordinates will be aligned in the low dimensional space by different local affine transformations to obtain a global coordinate system.\ref{143254}  
     c/- LTSA   : LOCAL TANGENT SPACE ALIGNMENT

 

           is a technique that describes local properties of the high-dimensional data using the local tangent space of each datapoint . LTSA is   algorithmically similar to  LLe can be put in this category. One advantage of LTSA over LLE is that using LTSA we can potentially detect the intrinsic dimension of the underlying manifold by analyzing the local tangent space structure.\ref{391140}  
The LTSA  algorithm comprises three stages:
  1.      Nearest Neighbors Search . Determine k nearest neighbors xij of xi, j = 1, . . . , k.
  2. Weight Matrix Construction( Constructing alingment matrix). Form the matrix B by locally summing (7.18) in the article if a direct eigen-solver will be used. Otherwise implement a routine that computes matrix-vector multiplication Bu for an arbitrary vector u.
  3. Aligning global coordinates Compute the d + 1 smallest eigenvectors of B and pick up the eigenvector matrix [u2, · · · , ud+1] corresponding to the 2nd to d+1st smallest eigenvalues, and set  T = [u2, · · · , ud+1]T .
for using LTSA in matlab we need to use function
function mappedX = ltsa(X, no_dims, k, eig_impl) with 
The function runs the local tangent space alignment algorithm on dataset  X, reducing the data to dimensionality d. The number of neighbors isspecified by k.