Consider a rank minimization problem over two positive semi-definite matrices $X$ and $Y$ with $R = \operatorname{rank}(X) = \operatorname{rank}(Y)$: \begin{equation*} \begin{aligned} & \underset{X,Y}{\text{min}} & & R\\ &\text{subject to}& & \| X - \hat{X} \| \leq \epsilon_{1}\\ &&& \| Y - \hat{Y} \| \leq \epsilon_{2}\\ &&& X,Y \succeq 0 \\ &&& R = \operatorname{rank}(X) = \operatorname{rank}(Y)\\ \end{aligned} \end{equation*}
I could potentially solve (locally) the problem using a variant of the alternating projection method by simultaneously minimizing the rank of both matrices in each iteration.
Since both X and Y are PSD, I was curious if you could solve the problem using the trace (nuclear norm) heuristic, by encoding the problem in terms of, say, a linear combination of traces of PSD matrices $X$ and $Y$, subject to the above constraints. But that doesn't seem to guarantee the equality of ranks!
Any other approaches or insights into this problem or pointers is appreciated.