Does it matter if the preconditioner maps a point on the tangent space to elsewhere?

[张亦弛]

Good day.

I am trying to design a preconditioner to accelerate the tCG convergence. If I have got it correct, a preconditioner is the approximate inverse operator of the Riemannian hessian operator. Both the preconditioner and the hessian operator should be maps from the tangent space to itself. But now I am confronted with a situation that the preconditioner maps from the tangent space to elsewhere (not the tangent space).

The problem originates from this. Although the hessian operator (Y=f(X)) is very straight forward (matrix multiplication and element-wise multiplication).The reverse way (X=f^-1(Y)) is not. The reverse way requires solving a matrix equation. Although this matrix equation is guaranteed to have solutions and they are already found, the solutions appear to be multiple, with a set of arbitrary coefficients. I guess that if the coefficients are wisely chosen, there must be a unique solution that lies on the tangent space. However, I do not think there is any easy approach to find the good coefficients, and all I can do is simply set the coefficients to zero. The good news is that the expression is simplified in this way; the bad news is that the resulting solution is not on the tangent space.

I hope the shortcoming of my preconditioner will not matter. After all, when the preconditioner is applied to both sides of the Riemannian Newton equation "hess f(p)[X] = grad f(p)", the equation should still hold, doesn't it? But I am not sure.

Thanks for any help. I can provide more details about the stuff I am working on if anyone is interested.

[Nicolas Boumal]

Hi,

For the usual theory to hold, the preconditioner at x indeed needs to be a positive definite map from the tangent space at x to the tangent space at x. If not, then algorithm could easily break and behave in unforeseen ways. That said, one can always try; and if things work well, then maybe it's possible to explain why after the fact.

Best,

Nicolas

[张亦弛]

Thank you for the explanation. It's prevented me from moving forward to an incorrect direction.