Laplacian Eigenmaps modification using adaptive graph for pattern recognition

Laplacian Eigenmaps (LE) is a typical nonlinear graph-based (manifold) dimensionality reduction (DR) method, applied to many practical problems such as pattern recognition and spectral clustering. It is generally difficult to assign appropriate values for the neighborhood size and heat kernel parameter for LE graph construction. In this paper, we modify graph construction by learning a graph in the neighborhood of a pre-specified one. Moreover, the pre-specified graph is treated as a noisy observation of the ideal one, and the square Frobenius divergence is used to measure their difference in the objective function. In this way, we obtain a simultaneous learning frame work for graph construction and projection optimization. As a result, we obtain a principled edge weight updating formula which naturally corresponds to classical heat kernel weights. Experimental result using UCI datasets and different classifiers show the feasibility and effectiveness of the proposed method in comparison to conventional LE for the classification.