Non-negative Laplacian Embedding

Laplacian embedding provides a low dimensional representation for a matrix of pairwise similarity data using the eigenvectors of the Laplacian matrix. The true power of Laplacian embedding is that it provides an approximation of the Ratio Cut clustering. However, Ratio Cut clustering requires the solution to be {\it nonnegative}. In this paper, we propose a new approach, nonnegative Laplacian embedding, which approximates Ratio Cut clustering in a more direct way than traditional approaches. From the solution of our approach, clustering structures can be read off directly. We also propose an efficient algorithm to optimize the objective function utilized in our approach. Empirical studies on many real world datasets show that our approach leads to more accurate Ratio Cut solution and improves clustering accuracy at the same time.