## Appendix A Proof of Theorem 1  
It is well known that the following inequalities hold at an optimum point [29]:  
\[J(\mathbf{P}^{\star},\mathbf{\tilde{T}},\mathbf{\tilde{R}},\mathbf{\tilde{Q}}) \leq J(\mathbf{P}^{\star},\mathbf{\tilde{T}}^{\star},\mathbf{\tilde{R}}^{\star },\mathbf{\tilde{Q}}^{\star})\leq J(\mathbf{P},\mathbf{\tilde{T}}^{\star}, \mathbf{\tilde{R}}^{\star},\mathbf{\tilde{Q}}^{\star}) \tag{11}\]  
for all feasible values of \(\mathbf{P}\), \(\mathbf{\tilde{T}}\), \(\mathbf{\tilde{R}}\), and \(\mathbf{\tilde{Q}}\), In other words, a saddle point of the MSE \(J\) in (12) is a solution to the worst-case MSE minimization problem in (14). Let us define the function \(\mathcal{L}\) as  
\[\mathcal{L}(\mathbf{P},\mathbf{\tilde{T}},\mathbf{\tilde{R}}, \mathbf{\tilde{Q}},\mu_{P},\mu_{T},\mu_{R},\mu_{Q},\mathbf{\Gamma}_{T}, \mathbf{\Gamma}_{R},\mathbf{\Gamma}_{Q})\] \[=J(\mathbf{P},\mathbf{\tilde{T}},\mathbf{\tilde{R}},\mathbf{ \tilde{Q}})+\mu_{P}\Big{[}\text{Tr}(\mathbf{P}\mathbf{P}^{H})-P_{T}\Big{]}\] \[\quad-\mu_{T}\Big{[}||\mathbf{\tilde{T}}||_{F}^{2}-e_{T}\Big{]}+ \text{Tr}\big{[}(\mathbf{\hat{T}}+\mathbf{\tilde{T}})\mathbf{\Gamma}_{T}\big{]}\] \[\quad-\mu_{R}\Big{[}||\mathbf{\tilde{R}}||_{F}^{2}-e_{R}\Big{]}+ \text{Tr}\big{[}(\mathbf{\hat{R}}+\mathbf{\tilde{R}})\mathbf{\Gamma}_{R}\big{]}\] \[\quad-\mu_{Q}\Big{[}||\mathbf{\tilde{Q}}||_{F}^{2}-e_{Q}\Big{]}+ \text{Tr}\big{[}(\mathbf{\hat{Q}}+\mathbf{\tilde{Q}})\mathbf{\Gamma}_{Q}\big{]}, \tag{12}\]  
where \(\mu_{P}\), \(\mu_{T}\), \(\mu_{R}\), \(\mu_{Q}\), \(\mathbf{\Gamma}_{T}\in\mathbb{C}^{M_{t}\times M_{t}}\), \(\mathbf{\Gamma}_{R}\in\mathbb{C}^{M_{t}\times M_{t}}\), and \(\mathbf{\Gamma}_{Q}\in\mathbb{C}^{L\times L}\) are dual variables associated with the constraints of the problem (14). Then, the saddle point should satisfy the following KKT optimality conditions [28; 29]:  
\[\mathbf{\
abla}_{\mathbf{P}}\mathcal{L}|_{\mathbf{P}=\mathbf{P}^{\star}}=- \mathbf{Z}^{-2}\mathbf{P}^{\star}(\mathbf{\hat{Q}}+\mathbf{\tilde{Q}}^{\star })^{-1}+\mu_{P}\mathbf{P}^{\star}=\mathbf{0}, \tag{13a}\] \[\mu_{P}\geq 0,\quad\mu_{P}[\text{Tr}(\mathbf{P}^{\star}\mathbf{P}^{ \star H})-P_{T}]=0,\] (13b) \[\mathbf{\
abla}_{\mathbf{\hat{T}}}\mathcal{L}|_{\mathbf{T}=\mathbf{\tilde{T}}^ {\star}}=-(\mathbf{\hat{T}}+\mathbf{\tilde{T}}^{\star})^{-1}\mathbf{Z}^{-2}( \mathbf{\hat{T}}+\mathbf{\tilde{T}}^{\star})^{-1}+\mu_{T}\mathbf{\tilde{T}}^ {\star}-\mathbf{\Gamma}_{T}=\mathbf{0},\] (13c) \[\mu_{T}\geq 0,\quad\mu_{T}\Big{[}||\mathbf{\tilde{T}}^{\star}||_{F}^{2 }-e_{T}\Big{]}=0,\] (13d) \[\mathcal{L}(\mathbf{P},\mathbf{\tilde{T}},\mathbf{\tilde{R}}, \mathbf{\tilde{Q}},\mu_{P},\mu_{T},\mu_{R},\mu_{Q},\mathbf{\Gamma}_{T}, \mathbf{\Gamma}_{R},\mathbf{\Gamma}_{Q})\]0 (13e) \[\mathcal{L}(\mathbf{P},\mathbf{\tilde{T}},\mathbf{\tilde{R}}, \mathbf{\tilde{Q}},\mu_{P},\mu_{T},\mu_{R},\mu_{Q},\mathbf{\Gamma}_{T}, \mathbf{\Gamma}_{R},\mathbf{\Gamma}_{Q})\]1 (13f) \[\mathcal{L}(\mathbf{P},\mathbf{\tilde{T}},\mathbf{\tilde{R}}, \mathbf{\tilde{Q}},\mu_{P},\mu_{T},\mu_{R},\mu_{Q},\mathbf{\Gamma}_{T}, \mathbf{\Gamma}_{R},\mathbf{\Gamma}_{Q})\]2 (13g) \[\mathcal{L}(\mathbf{P},\mathbf{\tilde{T}},\mathbf{\tilde{R}}, \mathbf{\tilde{Q}},\mu_{P},\mu_{T},\mu_{R},\mu_{Q},\mathbf{\Gamma}_{T}, \mathbf{\Gamma}_{R},\mathbf{\Gamma}_{Q})\]3 (13h) \[\mathcal{L}(\mathbf{P},\mathbf{\tilde{T}},\mathbf{\tilde{R}}, \mathbf{\tilde{Q}},\mu_{P},\mu_{T},\mu_{R},\mu_{Q},\mathbf{\Gamma}_{T}, \mathbf{\Gamma}_{R},\mathbf{\Gamma}_{Q})\]4 (13i) \[\mathcal{L}(\mathbf{P},\mathbf{\tilde{T}},\mathbf{\tilde{R}}, \mathbf{\tilde{Q}},\mu_{P},\mu_{T},\mu_{R},\mu_{Q},\mathbf{\Gamma}_{T}, \mathbf{\Gamma}_{R},\mathbf{\Gamma}_{Q})\]5 (13j) \[\mathcal{L}(\mathbf{P},\mathbf{\tilde{T}},\mathbf{\tilde{R}}, \mathbf{\tilde{Q}},\mu_{P},\mu_{T},\mu_{R},\mu_{Q},\mathbf{\Gamma}_{T}, \mathbf{\Gamma}_{R},\mathbf{\Gamma}_{Q})\]6  
where \(\mathbf{Z}=(\mathbf{\hat{T}}+\mathbf{\tilde{T}}^{\star})^{-1}+\mathbf{P}^{\star }(\mathbf{\hat{Q}}+\mathbf{\tilde{Q}}^{\star})^{-1}\mathbf{P}^{\star H}\).