Benchmarking against other libraries

[mateuszbaran]

I've started this to keep #333 closed but still discuss benchmarks.

[mateuszbaran]

First, Optim.jl and Manopt.jl have slightly different gradient tolerance criteria. Manopt has gradient 2-norm while Optim.jl uses inf-norm: https://julianlsolvers.github.io/Optim.jl/stable/#user/config/

[mateuszbaran]

For me the text is not completely clear, since it just says “Absolute tolerance in the gradient, in infinity norm” – tolerance to what? The two before speak about change, so from so similar names I concluded it might be gradient change - but nevertheless, also for just the gradient norm, the same problem exists, sure.

Judging by the results of Optim's state printing, it is gradient norm, at it doesn't even support norm-of-gradient-difference stopping condition.

What do you mean by “manifold-dependent”? I would say it is basis-dependent, which is a bit more than that. The Riemannian metric would be manifold dependent, but this one is additionally depending on the choice of basis.

Sure, it depends on both. I'm mostly thinking about the possibility of basis being implicit for stopping condition because I don't see much use of inf-norm beyond it being faster to compute and comparing to Optim.jl.

[kellertuer]

it is surely possible, though with the basis decomposition we additionally have to do only in rare cases faster.

[mateuszbaran]

That's exactly why I'm talking about implicit basis: to avoid basis decomposition.

[kellertuer]

This I do not follow. also for an implicit basis I would consider the map that transforms the tangent vector to its coefficients a decomposition. Optim.jl or 2-norm does not do this, so this is usually slow. Sure Euclidean with ONB (=decomp is the identity) there it should be fine ;)

[mateuszbaran]

I'm suggesting just calling norm(X, Inf) for all tangent vectors represented by arrays 😉, and adding a mathematical explanation why this is OK. It seems to be. There is a basis in which that is an Inf norm.

[kellertuer]

As a thread this is a great idea to collect and discuss these before opening (maybe more precise issues or) PRs! Thanks!