Multi-Scale Kernels Using Random Walks

We introduce novel multiscale kernels using the random walk framework and derive corresponding embeddings and pairwise distances. The fractional moments of the rate of continuous time random walk (equivalently diffusion rate) are used to discover higher order kernels (or similarities) between pair of points. The formulated kernels are isometry, scale, and tessellation invariant, can be made globally or locally shape aware, and are insensitive to partial objects and noise based on the moment and influence parameters. Additionally, the corresponding kernel distances and embeddings are convergent and efficiently computable. We introduce dual GMS signatures based on the kernels and discuss the applicability of the multiscale distance and embedding. Collectively, we present a unified view of popular embeddings and distance metrics while recovering intuitive probabilistic interpretations on discrete surface meshes.