From Laplacian Eigenmaps to Kernel Locality Preserving Projections: Equivalence or Improvement?

Abstract

Kernel Locality Preserving Projections (KLPP) and Laplacian eigenmaps (LE) are often taken as two different kinds of approaches in the application of nonlinear dimensionality reduction, but they are more closely related actually than expected. In this paper, KLPP is proved theoretically to solve exactly the same constrained minimization problem as LE. However, the application of KLPP is sensitive to the selections of kernel type and parameters, whereas LE is more efficient and straightforward. Unfolding results on different datasets of the two approaches are presented, together with the comparison of the computation time between KLPP and LE. In our experiments, the actual running time of LE is shorter than that of KLPP, though the time complexity of the two algorithms is comparable. The conclusion of this paper is a beneficial supplement to the nonlinear dimensionality reduction methods system and can be generalized to other algorithms.