Vertex weights for get_laplacian

[msseibel]

The current implementation of get_laplacian provides 3 types of normalization for the Laplacian (None, sym , rw).

For the following discussion I refer to Bronstein et al. "Geometric deep learning: going beyond Euclidean data" .

The None and rw normalization techniques can be derived from L = A⁻¹ (D - W).
D is the degree matrix: D = np.diag(np.sum(W,axis=1)) - np.diagonal(W).
A is the matrix of vertex weights.

For obtaining the unnormalized Laplacian (None) we set A = Identity.
To get rwwe set A = D.

Can we have a more general interface where we specify the matrix A?

[rusty1s]

We currently do not support the notion of defining vertex weights in get_laplacian, but I'm definitely open for a more general interface. Do you have a use-case where this might be important?

[msseibel]

Thank you for the fast reply.

I have an approach from traditional signal processing where the Laplacian for a manifold is defined as above.
In short: A signal (electric potential) f is sampled at vertices of a triangular mesh. The authors use the finite element method (FEM) and assume that f changes linearly from one vertex to another. The matrix A is a result from the FEM discretization. I can imagine that the FEM discretization is also useful for modelling other physical cases (e.g. heat transfer).

I wanted to show that geometric deep learning generalizes this approach.

Luckily the authors also use the cotangent weights for discretization where I don't need the matrix A. So I will stick with the simpler approach for now.