## 2 System Model
### Problem Formulation  
With the two-stage precoder and the quantized CDI of the effective channel, the signal-to-interference plus noise ratio (SINR) of user \(k\) in the \(g\)-th group is obtained from (5) as follows:  
\[\mathrm{SINR}_{g,k}=\frac{\gamma\left|\left(\mathbf{\mathrm{h}}_{g,k}^{ \mathrm{eff}}\right)^{\dagger}\mathbf{\mathrm{v}}_{g,k}\right|^{2}}{\gamma\underbrace {\sum_{i\
eq k}^{K_{g}}\left|\left(\mathbf{\mathrm{h}}_{g,k}^{\mathrm{eff}}\right) ^{\dagger}\mathbf{\mathrm{v}}_{g,i}\right|^{2}}_{\mathrm{SGI}}+\underbrace{\gamma \sum_{c\
eq g}^{G}\sum_{j=1}^{K_{c}}\left|\mathbf{\mathrm{h}}_{g,k}^{\dagger}\mathbf{ \mathrm{T}}_{c}\mathbf{\mathrm{v}}_{c,j}\right|^{2}}_{\mathrm{IGI}}+1}, \tag{23}\]  
where \(\gamma\triangleq\frac{p}{K\sigma^{2}}\) is the signal-to-noise ratio (SNR) at each user. Then, the average sum rate denoted by \(\mathcal{R}_{\mathrm{sum}}\) is given by  
\[\mathcal{R}_{\mathrm{sum}}=\sum_{g=1}^{G}\sum_{k=1}^{K_{g}}\mathbb{E}\left[ \log_{2}\left(1+\mathrm{SINR}_{g,k}\right)\right]. \tag{24}\]  
Analyzing (23) and (24), the quantized CDI of the effective channel, i.e., \(\mathbf{\mathrm{h}}_{g,k}^{\mathrm{eff}}\), is only fed back to the BS, so the IGI term in \(\mathrm{SINR}_{g,k}\) is not affected by the quantized CDI. In this case, the IGI term is determined by the outer precoders for the other groups, i.e., \(\{\mathbf{\mathrm{T}}_{c}\}_{c\
eq g}^{G}\). With exploiting the BD method for the outer precoder design as in (10), the performance of IGI cancellation, i.e., \(\sum_{c\
eq g}^{G}\sum_{j=1}^{K_{c}}\left|\mathbf{\mathrm{h}}_{g,k}^{\dagger}\mathbf{ \mathrm{T}}_{c}\mathbf{\mathrm{v}}_{c,j}\right|^{2}\), depends on the reduced dimension numbers of the effective channels of the other groups, i.e., \(\{s_{c}\}_{c\
eq g}^{G}\). On the other hand, the magnitude of the desired signal term, i.e., \(\left|\left(\mathbf{\mathrm{h}}_{g,k}^{\mathrm{eff}}\right)^{\dagger}\mathbf{\mathrm{v }}_{g,k}\right|^{2}\), and the performance of SGI cancellation, i.e., \(\sum_{i\
eq k}^{K_{g}}\left|\left(\mathbf{\mathrm{h}}_{g,k}^{\mathrm{eff}}\right) ^{\dagger}\mathbf{\mathrm{v}}_{g,i}\right|^{2}\), are restricted by the dimension of the outer precoder of \(g\)-th group, i.e., \(s_{g}\)[19]. Note that the quantization error of the effective channel, i.e., \(\hat{\mathbf{\mathrm{h}}}_{g,k}^{\mathrm{eff}}\) in (20), is more limited by \(s_{g}\) given feedback bit allocation, i.e., \(B\)-bits. Therefore, the reduced dimensions of the effective channels for all groups (i.e., \(s_{1},\ldots,s_{G}\)) should be jointly optimized considering all the interactions among them in order to maximize the average sum rate in (24). Thus, we formulate the optimization problem to find the optimal \(s_{1},\ldots,s_{G}\) as follows [19]:  
\[\mathbf{P1}: \;\;\underset{s_{1},\ldots,s_{G}}{\text{maximize}} \sum_{g=1}^{G}\sum_{k=1}^{K_{g}}\mathbb{E}\left[\log_{2}\left(1+ \mathrm{SINR}_{g,k}\right)\right],\] (25) subject to \[s_{g}\in\mathbb{Z}^{+},\;\forall g=1,..,G,\] \[K_{g}\leq s_{g}\leq r_{g}.\]The optimization problem \(\mathbf{P1}\) is difficult to solve directly because it is a mixed-integer problem, which is generally known to be NP-hard. Moreover, the effect of the dimensions is implicit in the objective function, and the optimal solution can be obtained only by numerical search for every channel realization, which is almost impossible in practice. Note that, once the reduced dimensions \(\{s_{g}\}_{g=1}^{G}\) are determined, the BS informs \(s_{g}\) to the users in the \(g\)-th group so that they can quantize \(s_{g}\)-dimensional effective channels with the corresponding codebook.