two orthogonal Stiefel manifolds

[nickolay.t...@open.ac.uk]

Hi,

I have a problem with two unknowns F and U from Stiefel(n,r) and Steifel(n,p). However, they should also fulfil F'U = 0.

Seems like Manopt cannot help directly (there is need of some more derivations). But, correct me if I'm wrong.

Thanks,

Nickolay

[BM]

Dear Nickolay,

Can't we have X = [F U]  and X is n-by-(r+p) on the Stiefel manifold (n, r + p)?

Regards,

Bamdev

[nickolay.t...@open.ac.uk]

Dear Bamdev,

Unfortunately not. The objective function cannot be re-written in terms of [F U].

I have a way out, by modifying the gradients. I guess that Manopt should do something similar.

All the best,

Nickolay

[BM]

The objective function need not use X directly. In our cost, we say F as X(:,1:r) and U as X(:,r+1:end) and define the cost. Am I missing something?

[BM]

sample pseudo code.

n = 20;

r = 5; 

p = 2;

problem.M = stiefelfactory(n, r+p);

problem.cost = @mycost;

function obj = mycost(X)

F = X(:,1:r);

U = X(:,r+1:end);

obj = some functiont of F and U.

end

problem.egrad = @myegrad;

function g = myegrad(X)

F = X(:,1:r);

U = X(:,r+1:end);

compute gradient wrt F, say, gradF. Similarly, compute gradient wrt U, say gradU.

g = [gradF gradU];

end