Extensions and Analysis of Local Non-linear Techniques

The techniques Conformal Eigenmap and Neighborhood Preserving Embedding (NPE) have been proposed as extensions of local non-linear techniques. Many of the commonly used non-linear dimensionality reduction, such as Local Linear Embedding (LLE) and Laplacian eigenmap are not explicitly designed to preserve local features such as distances or angles. In first proposed Conformal Eigenmap technique, a low dimensional embedding is constructed that maximally preserves angles between nearby data points. The embedding is derived from the bottom eigenvectors of LLE by solving an additional problem in Semidefinite Programming (SDP). In second proposed method, NPE minimizes the cost function of a local nonlinear technique for dimensionality reduction under the constraint that the mapping from the high-dimensional to the low-dimensional data representation is linear. The idea is to modify the LLE by introducing a linear transform matrix. The effectiveness of the proposed methods is demonstrated on synthetic datasets. Experimental results on several data sets demonstrate the merits of proposed techniques.