Working with the tangential space of a manifold

[volkerkarle]

Hi there, thanks for your amazing work!

I was wondering if I could use this approach to iteratively work out the tangential space of the manifold defined by f(x)=0, with x ∈ ℝⁿ, f ∈ℝᵈ. Right now I'm using the implicit function theorem to calculate the tangential vectors. However, with the naive implementation (using DiffOpt.jl) the accuracy is very low compared to the tangential vectors I get by sampling over points and taking x-x'/|x-x'| for x-x'->0. I do not know about the concrete details of your implementation, but I have the impression that it is superior to my naive Ansatz.

Further, do you think it could be possible to calculate the curvature tensor (and henceforth, the parallel transport) of the manifold at each point? In the end I'm interested in singularities (i.e. bifurcations) and for this I need to be able to calculate derivatives up to high precision.

Best,

v.

[gdalle]

Hi @volkerkarle! I'm not an expert on Manifolds, but I imagine you've already looked at Manifolds.jl without success?

[gdalle]

From what I can tell, if you use implicit differentiation on $f(x) = 0$, you will get a tangent vector, the direction of which is determined by the AD internals. I'm not sure how to get several

[volkerkarle]

Hi Guillaume,

So manifold.jl if you are working with already well defined manifolds. I did not see how one can apply it for a implicitly defined manifolds, but it's not an issue. As soon as the tangential space is (up to some precision) well defined, curvature etc. follows straightforwardly.
To get all (i.e. several) directions is also not an issue, it's sufficient to just get all derivatives and then compute a minimal basis for the tangent space (for instance, by SVD). My main concern is the accuracy of the derivatives. I cannot try it out now, but later in the month I'll give your package a try and see how it performs!

[gdalle]

As far as I know, implicit differentiation is exact, up to the precision of the linear solver.

[gdalle]

Hey @volkerkarle, did you end up succeeding in your endeavors?