Signed Laplacian Embedding for Supervised Dimension Reduction

Manifold learning is a powerful tool for solving nonlinear dimension reduction problems. By assuming that the high-dimensional data usually lie on a low-dimensional manifold, many algorithms have been proposed. However, most algorithms simply adopt the traditional graph Laplacian to encode the data locality, so the discriminative ability is limited and the embedding results are not always suitable for the subsequent classification. Instead, this paper deploys the signed graph Laplacian and proposes Signed Laplacian Embedding (SLE) for supervised dimension reduction. By exploring the label information, SLE comprehensively transfers the discrimination carried by the original data to the embedded low-dimensional space. Without perturbing the discrimination structure, SLE also retains the locality. Theoretically, we prove the immersion property by computing the rank of projection, and relate SLE to existing algorithms in the frame of patch alignment. Thorough empirical studies on synthetic and real datasets demonstrate the effectiveness of SLE.