Implementation of kernels on the edge space of graphs or simplicial complexes

[cookbook-ms]

This pull requests an implementation of kernels on the Edge space of a graph or a simplicial 2-complex. I implemented the following:

1. the discrete space of edge space: written as a class in geometric_kernels/spaces/graph_edge.py, which allows backends based on numpy, pytorch and jax.
2. tutorials on this, as how graph kernel tutorial looks like. This includes:

• numpy backend:

  an example of the edge space of a graph, written in jupyter notebook in notebooks/EdgeSpaceGraph.ipynb an example of the edge space of a simplicial 2-complex, written in notebooks/EdgeSpaceSimplicialComplex

• jax and pytorch backends:

  an example of the edge space of a graph, written in notebooks/backends/Jax_EdgeSpaceGraph.ipynb and notebooks/backends/PyTorch_EdgeSpaceGraph.ipynb an example of the edge space of a simplicial 2-complex, written in notebooks/backends/Jax_EdgeSpaceSimplicialComplex and notebooks/backends/PyTorch_EdgeSpaceSimplicialComplex

3. testing on the space of GraphEdge: written in "tests/spaces/test_graph_edge.py"

Note that:

• The above 8 files are the ones I wrote. All the modifications elsewhere are only about that I added few lines to look for the correct path of the geometric_kernels module (without installing it, since I need to test my own GraphEdge space).
• The current implementation in fact does not contain Hodge-compositional kernels, instead, they are using the one set of kernel parameters on the whole eigenspace. To implement the Hodge-compositional kernels, we can define the three sets of eigenspaces individually, then allowing different sets of parameters.

[cookbook-ms]

Sorry for the late updates.

In this update, we have the following changes:

1. the space GraphEdge takes a graph and optional triangle list as inputs and builds a simplicial 2-complex. If no triangle list is provided, we look for all the 2-cliques. Of course, one can choose only using graph instead of simplicial 2-complex.
2. the space GraphEdge gives an eigenbasis together with their Hodge type specified, so to enable Hodge-compositional kernel
3. a Hodge-compositional kernel is enabled in a separate class MaternKarhunenLoeveKernel_HodgeCompositionEdge defined in geometric_kernels/kernels/karhunen_loeve.py which takes parameters for three subspaces as inputs.
4. minimal changes are made in geometric_kernels/kernels/matern_kernel.py, as well as in geometric_kernels/feature_maps/deterministic.py and geometric_kernels/sampling/samplers.py, to enable the option of using Hodge-compositional kernel.
5. tutorials in numpy, jax and pytorch are provided in notebooks/SimplicialComplex.ipynb, notebooks/backends/JAX_SimplicialComplex.ipynb, and notebooks/backends/PyTorch_SimplicialComplex.ipynb which give how to define Hodge-compositional edge kernels in a simplicial 2-complex, as well as Hodge feature maps.

Please have a look at your convenience. Thanks :)

[vabor112]

Thanks a lot, Maosheng! I plan to take a look in a week or two, after the deadline streak has passed.