Sparsity support in pytensor

[Ch0ronomato]

Description

I'm investing implementing ALS in pytensor which is usually implemented with sparsity constructs (see implicit for reference). I quickly looked around and saw this older thread where someone asked for sparsity support. @jessegrabowski gave a first pass answer, but mentioned the support is subpar. Opening this to track any enhancements we could bring.

[jessegrabowski]

Is this specifically about implementing solve with sparse inputs, or do you have other Ops in mind?

[Ch0ronomato]

The original issue seems to be just solve, I imagine that's good enough to start

[jessegrabowski]

So for solve it's easy enough to wrap sparse.linalg.spsolve for the C backend. We need gradients, but I found a JAX implementation here that we can copy, so that should be easy enough.

For the numba backend we need to write our own overrides, as I did for solve_discrete_are, for example. spsolve calls out to superlu, which I think we can hook into the same way we do for the other cLAPACK functions. There's also another optional package umfpack which scipy appears to prefer if it's available, but it would be a new dependency.

Finally for the Torch backend I honestly have no idea. It looks like torch has sparse support as well as an spsolve implementation, so it might be straight forward?

[Ch0ronomato]

Great, thanks for the plan!

Out of curiosity, what could we do? I'm not aware of what general sparsity support would look like beyond making sure things can use csr and csc tensors, which i think pytensor already has sparse for that

[jessegrabowski]

Some thoughts:

1. Agreement/harmonization on the status of the SparseMatrix primitive. I thought there was an issue or discussion about this, but for the life of me I can't find it @ricardoV94 .
2. Support for as full a suite of linear algebra operations on sparse matrices as possible. This page has the list of what I would consider to be possible. Interesting ones would be spsolve, spsolve_triangular, eigs, svds, and kron.
3. Better rewrites/detection for when sparse matrices are involved in Ops, for example BUG: .dot() not producing the same as sparse.basic.dot() #321
4. Support for batch dimensions, see Allow sparse variables to have dummy dimensions to the left? #839
5. Support for symbolic sparse jacobians from pytensor.gradient.jacobian, see https://github.com/mfschubert/sparsejac