Needed Functionalities

[meyerscetbon]

• Continuous discrete Barycenter
• Scaling mechanism for Geometry Object, relying for instance on maximal norms of points
• Initialization for LRSinkhorn using rank-2 matrices as in https://arxiv.org/abs/2103.04737

[marcocuturi]

All done! Thanks for your contributions!

[marcocuturi]

Just to give an update on what the plan is in following weeks:

• more implementations of existing Wasserstein barycenter algorithms, notably for generic Riemannian manifolds (as long as a cost operator has a barycenter operation that is coded). As an example of this, maybe consider geometry on the sphere / torus / psd matrices.
• debug the debiased continuous W barycenter (currently this is not correct)
• implementation of a discrete Gromov-Wasserstein barycenter, building e.g. on this for instance. In that process we might have to refresh the discrete barycenter approach.
• implementation of a continuous Gromov-Wasserstein barycenter.
• automatic re-normalization approaches for costs that are selected by default in the solvers.

if you need another functionality for your project, please do not hesitate to add it here!

[zwei-beiner]

Hi, are Wasserstein/sinkhorn distances implemented for generic manifolds (in particular, the sphere)? Otherwise, are there any plans to implement this in the near future?

[marcocuturi]

OTT's philosophy is (https://ott-jax.readthedocs.io/en/latest/)

Split geometry from OT solvers in the discrete case: We argue that there should be one, and one implementation only, of every major OT algorithm (Sinkhorn, Gromov-Wasserstein, barycenters, etc…), regardless of the geometric setup that is considered. To give a concrete example, any speedups one may benefit from by using a specific cost (e.g. Sinkhorn being faster when run on a separable cost on histograms supported on a separable grid [Solomon et al., 2015]) should not require a separate reimplementation of a Sinkhorn routine.

If you only need to run sinkhorn on a geometric manifold that has a specific cost, you only need to implement a custom cost function for that manifold, as briefly explored in this tutorial where a custom cost is defined. There the custom cost is translation invariant (a function of x-y) but you could try to implement your own cost directly by simply overriding the pairwise(x,y) cost function.