Hessian sparse coding

Sparse coding has received an increasing amount of interest in recent years. It finds a basis set that captures high-level semantics in the data and learns sparse coordinates in terms of the basis set. However, most of the existing approaches fail to consider the geometrical structure of the data space. Recently, a graph regularized sparse coding (GraphSC) is proposed to learn the sparse representations that explicitly take into account the local manifold structure, which used graph Laplacian as a smooth operator. However, the GraphSC based on graph Laplacian suffers from the fact that sparse coordinates are biased toward a constant and the Laplacian embedding often cannot preserve local topology well as we expected. In this paper, we propose a novel sparse coding algorithm called Hessian sparse coding (HessianSC). HessianSC is based on the second-order Hessian energy, which favors functions whose values vary linearly with respect to geodesic distance. HessianSC can overcome the drawbacks of Laplacian based methods. We show that our algorithm results in significantly improved performance when applied to image clustering task.