Given a p.s.d kernel $Q$, can minimization/maximization of $Tr(X^TQX)$ over X be represented within a reproducing kernel Hilbert space (RKHS) framework? If there is a primary concern with the trace function being unbounded(for maximization) or trivial zero-solution(minimization); you may consider constraints over $X$; for example-say $X$ being orthogonal. Am trying to see if I can fit this in an RKHS framework.
Also, a few thoughts around this function using hilbert-schmidt norms are as follows: I do see that $TrX^TQX$ can be represented as $Tr[(SX)^T(SX)]=||SX||^2_{HS}$, using the hilbert schmidt norm where $S$ is the p.s.d square root of Q. (Ex: $S=U\lambda^{1/2}$, where $Q=U\lambda U^T$ is the eigen-decomposition of $Q$).
