Hessian of product manifold

[Evelyn LI]

Hi,

I am using the Manopt these days, and I find it really useful. 

I have some points which make me confused. In my case, a product manifold of complex Euclidean space and complex circles, I calculated the Euclidean gradients of two variables separately.  If I want to confirm the natural of the critical points,  how to derive the Hessian?  Calculating the Hessian separately or combining the two sets of variables together?

Thank you for your help. I am looking forward for your reply.  

Best,

Evelyn Li

[BM]

Hello,

For the Hessian computation, we should take the variables together.

Regards,
BM

[Evelyn LI]

Hi BM,

Thanks for your quick reply. 

I got it. First, we compute the Euclidean Hessian taking the two variables together. Then we project the Euclidean Hessian to the product manifold, right?

Thanks,

Evelyn 

[BM]

Hello Evelyn,

You are correct. Indeed the Manopt egrad2rgrad operation does that internally. We need to only supply the Euclidean Hessian in Manopt.

Regards,

BM