Many matrix variables

[bahu....@gmail.com]

I want to minimize a real valued function of n matrices V_i (each N x d with d<N) under orthogonality constraints V_i'*V_i=Id(d) for all i. How is this possible with Manopt?

[BM]

Hello,

You can use the powermanifold tool, details in http://manopt.org/tutorial.html#tools. 

For your specific case, you can follow the following steps.

M = stiefelfactory(N, d); % For each V_i

Mn = powermanifold(M, n); % n times repetition of M

An element of Mn is represented as a cell of length n.

Regards,

BM

[bahu....@gmail.com]

Thanks!

[Nicolas Boumal]

A more efficient way is to use the stiefelfactory, like so:

manifold = stiefelfactory(N, d, n);

Your points will be represented as 3D arrays (3D matrices) of size N x d x n, each slice being orthonormal. Tangent vectors are also represented as 3D matrices of the same size.

Cheers,

Nicolas

[bahu....@gmail.com]

Thanks! Thus if n=2, then this might be a typical cost function:
problem.cost = @(X) -trace(X(:,:,1)'*A1*X(:,:,1)+X(:,:,2)'*A2*X(:,:,2));
How would the gradient and the Hessian look like?

[BM]

Hello,

A point to note here is that there is symmetry in your cost function. For example, the transformation X(:, :, 1) -----> X(:, :, 1)*Q does not change the cost function for all orthogonal matrices of d-by-d, Q'*Q = Q*Q' = I.  Similarly for X(:, :, 2). Therefore, the optimization should be on the product of the Grassmann manifolds and not on the product of the Stiefel manifolds.

Attached is the code for your cost function.

Regards,
BM

[bahu....@gmail.com]

This is great! The code will be of help not only to me. Thanks!