centred Stiefel factory?

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

Hi,

I need orthonormal n x p matrices Q (Q'Q = I_p), which columns are centred, i.e. Q'1_n = 0_m.

Is there a way to achieve this with the current factories?

Thanks,

Nickolay

[Nicolas Boumal]

Hello Nickolay,

We do not have a factory for this at the moment. We have a factory for the Stiefel manifold of course, and we also have a factory for the subspace of centered matrices (https://github.com/NicolasBoumal/manopt/blob/master/manopt/manifolds/euclidean/centeredmatrixfactory.m) but not for the intersection of the two (is it clear that the intersection is a manifold?)

Perhaps by taking inspiration from both existing factories you could implement your own for the intersection? If so and if you are happy with its workings, we could integrate it to the toolbox.

(Btw, did you mean that the rows are centered, rather than the columns?)

Best,

Nicolas

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

Hi again,

If you define in stiefelfactory:

M.dim = @() k*(n*p - .5*p*(p+1) - p);

center = @(X) bsxfun(@minus, X, mean(X));

and change the retraction to

[Q, R] = qr(center(Y(:, :, i)), 0);

seems enough.

All the best,

Nickolay

[Nicolas Boumal]

Hello Nickolay,

I'm not sure I see how this works. If I have a matrix whose columns are orthonormal, let's call them v_1, ..., v_m, then it's not possible for these columns to be centered, as otherwise v_1 +  ... + v_m = 0, which would imply that the columns are linearly dependent (impossible if they are orthonormal).

Practically, in the proposed retraction [Q, R] = qr(center(Y(:, :, i)), 0); I can't imagine that centering subsides after taking the QR, right?

Best,

Nicolas