Add new initializer: hyper_init

[sm2909]

Fixes #303

Checklist

• Appropriate tests were added
• Any code changes were done in a way that does not break public API
• All documentation related to code changes were updated
• The new code follows the contributor guidelines, in particular the SciML Style Guide and COLPRAC.
• Any new documentation only uses public API

Implementation Details

The hyper_init function executes the following steps:

1. Sampling: Samples $N$ points on a Poincaré ball.
2. Distance Calculation: Calculates hyperbolic distances between all sampled points, populating an $N \times N$ kernel.
3. Weighting: Applies $\sigma$ on the kernel to establish geometry-aware kernel weights.
4. Sparsification: Sparsifies the kernel by selecting the top $k$ values in each row.
5. Scaling: Applies spectral scaling to finalize the matrix.

Refactoring & Helper Functions

To keep hyper_init clean, the logic is modularized into three helper functions (taking inspiration from the original Proof of Concept):

• hyperbolic_distance_poincare
• sample_poincare_points
• topk_row_sparsify

Mathematical Assumptions

A simplifying assumption is used when sampling the radius $\rho$ from the Poincaré ball. To match the approximation used in the PoC, the radius is sampled linearly:

rho = rand(rng) * R_max

Instead of the exact inverse transform sampling method:

u = rand(rng)
rho = acosh(u * (cosh(R_max) - 1) + 1)

Questions:

1. Method Signature: Could you please take a look at the method signature? If the kwargs need more clarity or better naming conventions, I am happy to adjust them.
2. Verification & Testing: How can I formally verify that the initialization values produced are mathematically accurate for a hyperbolic init? The procedural implementation matches the PoC from my understanding. If there is a method to test this, I am happy to test it.

[MartinuzziFrancesco]

remove #

[MartinuzziFrancesco]

better to avoid single letter kwargs

[MartinuzziFrancesco]

Thanks for looking into this! The implementation seems to be ok, although I will need to double check against the python code. There are a couple of minor changes to tackle in the meantime, I left some comments