the optimizer to the resulting optimization problem, then the optimizer to (\[formu:problem\_j\]) is given by $\sbra{\m{Z}^*\m{Q}_1^H,\m{u}^*}$. The same result holds for the nonconvex optimization problem in (\[formu:criterion3\]). \[prop:dimreduce\]

See Appendix \[sec:Append\_dimreduce\].

If only the frequencies are of interest, e.g., in DOA estimation, we can replace $\m{Y}_{\m{\Omega}}^{o}\m{Q}_1$ by $\sbra{\m{Y}_{\m{\Omega}}^{o}\m{Y}_{\m{\Omega}}^{oH}}^{\frac{1}{2}}$ (in fact, any matrix $\widetilde{\m{Y}}$ satisfying that $\widetilde{\m{Y}}\widetilde{\m{Y}}^H=\m{Y}_{\m{\Omega}}^{o}\m{Y}_{\m{\Omega}}^{oH}$) to obtain the same solution of $\m{u}$. \[rem:sameusolution\]

With a similar proof, the dimension reduction technique in Proposition \[prop:dimreduce\] can be extended to a more general linear model with observations expressed by $\m{\Phi}\m{Y}^o$ + noise, where $\m{\Phi}$ denotes a sensing matrix. Also, it can be applied to conventional discrete dictionary models, in which, for example, the $\ell_{2,p}$ norm, $0\leq p\leq1$, needs to be optimized. Note that the dimension reduction approach introduced here is different from that in [@malioutov2005sparse]. In particular, the approach in [@malioutov2005sparse] requires the knowledge of the model order $K$, which is unknown in practical scenarios, and gives an optimization problem which is an approximation of the original one. In contrast, our approach does not need $K$ but produces a dimension-reduced, equivalent problem.

By Proposition \[prop:dimreduce\], when $L>M$ we can reduce the order of the PSD matrix in (\[formu:weightedSDP\]) \[and (\[formu:duality\]), (\[formu:weightedSDP2\])\] 