The problem is as follows: $\displaystyle\min_{V}$ trace($V^TH^T\Phi HV$)$\\$ s.t. $V^TV=I_d$ in the case when $H$ is not known.
When $H$ is known, the solution is given by the eigenvectors corresponding to $d$ minimum eigenvalues of $ H^T\Phi H$.
Is there anyway to solve the problem in the absence of $H$? $H$ is a full column rank matrix. $\Phi$ is symmetric and positive semi-definite.
I hope to receive the suggestions if there are any. Or is there any technique that I can rephrase the problem.
Thank you for your help.
I think your objective function doesn't depend on your variable $V$ because we have : $\mbox{Trace}(AB)=\mbox{Trace}(BA)$ so in your problem we have : $\mbox{Trace}(V^TH^TQHV)=\mbox{Trace}(H^TQHVV^T)=\mbox{Trace}(H^TQHI)=\mbox{Trace}(H^TQH)$