## 4 Multi-Antenna Frequency-Selective Channels
### Transfer Learning for LSTD-Based Prediction  
Similar to conventional learning, transfer learning for LSTD-based prediction can be addressed from the naive extension (21) by utilizing the LSTD parametrization \(V^{(K)}\) in (34) in lieu of the unconstrained predictor \(V\) to obtain the bias matrix \(\bar{V}^{(K),\text{trans}}\) as  
\[\bar{V}^{(K),\text{trans}}=\operatorname*{arg\,min}_{\begin{subarray}{c}\hat{B},\hat{v}^{1},\ldots,\hat{v}^{K}\\ V^{(K)}=\sum\limits_{k=1}^{K}v^{k}\otimes(\hat{p}^{k}(\hat{p}^{k})^{\dagger}) \end{subarray}}\quad\quad\Big{\{}\sum\limits_{f=1}^{F}||X_{f}V^{(K)}-Y_{f}|| _{F}^{2}\Big{\}},\]  
which can also be solved via the ALS-based sequential approach detailed in Section 4.4. This produces the sequences \(\bar{b}^{1,\text{trans}},\ldots,\bar{b}^{K,\text{trans}}\) and \(\bar{v}^{1,\text{trans}},\ldots,\bar{v}^{K,\text{trans}}\) as (cf. (38))  
\[\bar{b}^{k,\text{trans}},\bar{v}^{k,\text{trans}}=\operatorname*{arg\,min}_{ \begin{subarray}{c}\hat{b}^{k},\hat{v}^{k}\\ (V^{(K)})^{k}=v^{k}\otimes(\hat{p}^{k}(\hat{p}^{k})^{\dagger})\end{subarray}} \quad\Big{\{}\sum\limits_{f=1}^{F}||X_{f}(V^{(K)})^{k}-(Y_{f})^{k}||_{F}^{2} \Big{\}},\]  
where the residual target matrix \((Y_{f})^{k}\) is defined as (cf. (39))  
\[(Y_{f})^{k}=\begin{cases}Y_{f},\text{ for }k=1,\\ Y_{f}-\sum\limits_{k^{\prime}=1}^{k-1}X_{f}(V_{f}^{(K)})^{k^{\prime},\text{ trans}},\text{ for }k>1\end{cases} \tag{46}\]  
with \(k\)-th optimized predictor  
\[(V^{(K)})^{k,\text{trans}}=\bar{v}^{k,\text{trans}}\otimes(\bar{b}^{k,\text{ trans}}(\bar{b}^{k,\text{trans}}))^{\dagger}. \tag{47}\]  
Details for transfer learning can be found in Appendix C, and the overall transfer learning scheme for LSTD prediction is summarized in Algorithm 2. After transfer learning, similar to Section 3.2, based on the optimized hyperparameters \(\bar{b}^{1,\text{trans}},\ldots,\bar{b}^{K,\text{trans}}\) and \(\bar{v}^{1,\text{trans}},\ldots,\bar{v}^{K,\text{trans}}\), the LSTD-based channel predictor for a new frame \(f^{\text{new}}\) can be obtained via (37) as  
\[V_{f^{\text{new}}}^{(K),*}=V^{(K),*}\big{(}\mathcal{Z}_{f^{\text{new}}}^{ \text{tr}}|\{\bar{b}^{k,\text{trans}},\bar{v}^{k,\text{trans}}\}_{k=1}^{K} \big{)}, \tag{48}\]  
which can also be solved in the sequential way as in (38).
```
Input: previous data \(\{\{h_{l,f}\}_{l=1}^{L+N+\delta-1}\}_{f=1}^{F}\); window size \(N\); prediction lag size \(\delta\); number of total features \(K\) Output: transfer-learned hyperparameters \(\bar{b}^{1,\text{trans}},\ldots,\bar{b}^{K,\text{trans}},\bar{v}^{1,\text{ trans}},\ldots,\bar{v}^{K,\text{trans}}\)
```
**Algorithm 2** LSTD-based transfer-learning for channel prediction for \(S\geq 1\)
```
Input: previous data \(\{\{h_{l,f}\}_{l=1}^{L+N+\delta-1}\}_{f=1}^{F}\); window size \(N\); prediction lag size \(\delta\); number of total features \(K\) Output: transfer-learned hyperparameters \(\bar{b}^{1,\text{trans}},\ldots,\bar{b}^{K,\text{trans}},\bar{v}^{1,\text{ trans}},\ldots,\bar{v}^{K,\text{trans}}\)
```
**Algorithm 3** LSTD-based transfer-learning for LSTD-based prediction for \(S\geq 1\)
```
Input: new frame \(f\) \(\{h_{l,f}\}_{l=1}^{L+N+\delta-1}\); window size \(N\); prediction lag size \(\delta\); number of total features \(K\) Output: transfer-learned hyperparameters \(\bar{b}^{1,\text{trans}},\ldots,\bar{b}^{K,\text{trans}},\bar{v}^{1,\text{ trans}},\ldots,\bar{v}^{K,\text{trans}}\)
```
**Algorithm 4** LSTD-based transfer-learning for LSTD-based prediction for LSTD-based