Compute discrete laplacian operator from distance matrix

Question

I have the $n$ by $n$ distance matrix for a set of $n$ (irregularly spaced) points. How can I compute the discrete Laplacian operator for these data?

For context, my distances are true geodesics across an irregular triangular 3d mesh surface.

Answer 1

Maybe you should read the article (apart from checking out the wikipedia page): Discrete Laplace Operators

Since you are working with discrete geometric notions, you should not rely purely on combinatorial descriptions of Laplace operators on graphs. You should implement discrete Laplacians that incorporate geometric information, not just combinatorial. Since, from what I have witnessed, no notion of discrete Laplacian captures all desired properties of classical Laplacians, you need to decide which properties are important for you and which are not. For that purpose however, you may need to calculate some more relevant geometric information from the geometry and combinatorics of your triangulated surface (Delaunay triangulations, Voronoy partitions, cotangents of angles of triangles...).

Answer 2

You cannot do so from a discrete set of points, because there are many triangulations that come from the same point set, and the discrete laplacian operator depends on the triangulation as well as the point set.

To make that more concrete, by choosing which diagonals to draw, you can triangulate the unit cube in 3-space so that $(0,0,0)$ and $(1,1,1)$ both have valence 6 or both have valence 3. The discrete laplacians, in these two cases, are different.