Exponential and Logarithm Mapping on Stiefel Manifold

[Ana Hiza Ramirez-andrade]

Hello,

I've trying to use manopt to solve a cost function of the form:

with x, the vectorization of X, an X matrix in the Stiefel Manifold. The mapping between the manifold and the tangent space is exponential. 

for this function, the gradient in the Stiefel Manifold is a 3x1 matrix, and the hessian is 3x3. 

I tried using the next lines, but I get an error due to the size of the gradient

M = stiefelfactory(3, 3);

problem.M = M;

problem.costgrad  = @(X)fandGrad(X,A,b,c);

problem.hess = @(X,u)Hessf(X,u,store);

[x_opt cost_opt info] = trustregions(problem, X0, options);

Is this the correct way to use the toolbox?
Do you have any suggestions to overcome this problem?

Thank you in advance.

Ana Ramirez 

[Nicolas Boumal]

Hello Ana,

I'm not clear on the setup. Is $r$ in $f(r)$ supposed to be $x$? Even so, I do not see what $\eta$ is. Looking at the expression for $\eta$ I'm wondering if what you actually want to optimize over is the orthogonal group?

In any case, M = stiefelmanifold(3, 3) will give you a structure M that describes the following manifold:

${ X \in R^{3  x 3}  :  X^T X = I_3 }$

where $I_3$ is the identity matrix of size 3 x 3.

So, points and vectors on the manifold are matrices of size 3 x 3. They are not column vectors. You can use Matlab instructions to vectorize them of course, but that is just a Matlab language issue.

Best,

Nicolas