Approximating Gradients for Meshes and Point Clouds via Diffusion Metric

The gradient of a function defined on a manifold is perhaps one of the most important differential objects in data analysis. Most often in practice, the input function is available only at discrete points sampled from the underlying manifold, and the manifold is approximated by either a mesh or simply a point cloud. While many methods exist for computing gradients of a function defined over a mesh, computing and simplifying gradients and related quantities such as critical points, of a function from a point cloud is non-trivial. In this paper, we initiate the investigation of computing gradients under a different metric on the manifold from the original natural metric induced from the ambient space. Specifically, we map the input manifold to the eigenspace spanned by its Laplacian eigenfunctions, and consider the so-called diffusion distance metric associated with it. We show the relation of gradient under this metric with that under the original metric. It turns out that once the Laplace operator is constructed, it is easier to approximate gradients in the eigenspace for discrete inputs (especially point clouds) and it is robust to noises in the input function and in the underlying manifold. More importantly, we can easily smooth the gradient field at different scales within this eigenspace framework. We demonstrate the use of our new eigen-gradients with two applications: approximating / simplifying the critical points of a function, and the Jacobi sets of two input functions (which describe the correlation between these two functions), from point clouds data.