## 4 Our Previous Work on Dimension Optimization  
As we explained in the previous section, the problem \(\mathbf{P1}\) is hard to solve because it is an NP-hard problem and the effect of the dimensions is implicit in the objective function. In this section, we briefly explain how we solved the problem \(\mathbf{P1}\) in our previous work [19]. In [19], we showed that the objective function of the problem \(\mathbf{P1}\) can be approximated [3] and then lower bounded as follows:  
\[\sum_{g=1}^{G}\sum_{k=1}^{K_{g}}\mathbb{E}\left[\log_{2}\left(1+ \text{SINR}_{g,k}\right)\right]\] \[\approx\sum_{g=1}^{G}\sum_{k=1}^{K_{g}}\log_{2}\left(1+\frac{ \mathbb{E}\left[\gamma\left|\left(\mathbf{h}_{g,k}^{\text{eff}}\right)^{ \dagger}\mathbf{v}_{g,k}\right|^{2}\right]}{\mathbb{E}\left[\gamma\sum_{i\
eq k }^{K_{g}}\left|\left(\mathbf{h}_{g,k}^{\text{eff}}\right)^{\dagger}\mathbf{v }_{g,i}\right|^{2}+\gamma\sum_{c\
eq g}^{G}\sum_{j=1}^{K_{c}}\left|\mathbf{h}_ {g,k}^{\dagger}\mathbf{T}_{c}\mathbf{v}_{c,j}\right|^{2}+1\right]}\right) \tag{26}\] \[>\sum_{g=1}^{G}\mathcal{R}_{g}^{\prime}\left(s_{1},\ldots,s_{G} \right), \tag{27}\]  
where  
\[\mathcal{R}_{g}^{\prime}\left(s_{1},\ldots,s_{G}\right)\triangleq\log_{2} \left(1+\frac{\gamma\left\{\text{Tr}\left(\mathbf{T}_{g}^{\dagger}\mathbf{R}_ {g}\mathbf{T}_{g}\right)-\sum_{j=1}^{K_{g}-1}\lambda_{j}\left(\mathbf{R}_{g} \right)\right\}}{\gamma\frac{K_{g}-1}{s_{g}-1}2^{-\frac{B}{g-1}}\text{Tr}\left( \mathbf{T}_{g}^{\dagger}\mathbf{R}_{g}\mathbf{T}_{g}\right)+\gamma\sum_{c \
eq g}^{G}K_{c}\text{Tr}\left(\mathbf{T}_{c}^{\dagger}\mathbf{R}_{g}\mathbf{ T}_{c}\right)+1}\right), \tag{28}\]  
with \(\lambda_{j}\left(\mathbf{R}_{g}\right)\) the \(j\)-th largest eigenvalue of \(\mathbf{R}_{g}\). Thus, the effect of dimensions becomes explicit in (27). Note that, given \(\{s_{g}\}_{g=1}^{G}\), the BS obtains the lower bound of (27) based only on the covariance matrices \(\{\mathbf{R}_{g}\}_{g=1}^{G}\) and the outer precoders \(\{\mathbf{T}_{g}\}_{g=1}^{G}\) without the CDI of \(s_{g}\)-dimensional effective channels (i.e., \(\mathbf{h}_{g,k}^{\text{eff}}\) for all \(g\in\{1,\ldots,G\}\) and \(k\in\{1,\ldots,K_{g}\}\)). For a practical design, we can establish using the lower bound (28) as follows:  
\[\mathbf{P2}:\ \underset{s_{1},\ldots,s_{G}}{\text{maximize}} \sum_{g=1}^{G}\mathcal{R}_{g}^{\prime}\left(s_{1},\ldots,s_{G} \right),\] (29) subject to \[s_{g}\in\mathbb{Z}^{+},\ \forall g=1,..,G,\] \[K_{g}\leq s_{g}\leq r_{g}.\]  
The problem \(\mathbf{P2}\) is also a mixed-integer problem, and obtaining the objective function of (29) in a closed-form is not available because the outer precoders \(\{\mathbf{T}_{g}\}_{g=1}^{G}\) are constructed by the procedure in (7)-(10) with respect to \(\{s_{g}\}_{g=1}^{G}\). Thus, the optimal solution to problem \(\mathbf{P2}\) is obtained by a combinatorial joint optimization of \(s_{1},\ldots,s_{G}\), which requires the \(G\)-dimensional numerical search, so it is still complex. To reduce complexity of the \(G\)-dimensional numerical search in problem **P2**, we can assume that the dimensions of the effective channels among all groups are the same as \(s\), i.e., \(s_{1}=\cdots=s_{G}=s\). Then, the problem **P2** is reduced to  
\[\textbf{P3}:\ \underset{s}{\text{maximize}} \sum_{g=1}^{G}\mathcal{R}_{g}^{\prime}\left(s\right),\] (30) subject to \[s\in\mathbb{Z}^{+}\] \[\max\{K_{1},...,K_{g}\}\leq s\leq\min\{r_{1},...,r_{G}\},\]  
and the optimal solution of the problem **P3** can be obtained from one-dimensional numerical search [19]. We describe the detailed procedure to optimal the optimal solution of the problem **P3** in Algorithm 1.