High-Order Regularization on Graphs

The Laplace-Beltrami operator for graphs has been been widely used in many machine learning issues, such as spectral clustering and transductive inference. Functions on the nodes of a graph with vanishing Laplacian are called harmonic functions. In differential geometry, the Laplace-de Rham operator generalizes the Laplace-Beltrami operator. It is a differential operator on the exterior algebra of a differentiable manifold, and it is equivalent to the Laplace-Beltrami operator when acting on a scalar function. In this paper, we develop a discrete analogue of the Laplace-de Rham operator, which naturally generalizes the discrete Laplace-Beltrami operator. The discrete Laplace-de Rham operator can then be used to define harmonic functions on arbitrary paths in a graph, in particular, functions on edges. Consequently, we build discrete regularization using the discrete Laplace-de Rham operator, and validate it on real-world web categorization tasks.