## 4 Multi-Antenna Frequency-Selective Channels
### Conventional Learning for LSTD-Based Prediction  
In conventional learning, the goal is to optimize the LSTD-based predictor \(V_{f}^{(K)}\) by optimizing the feature matrix \(\hat{B}_{f}\) and the feature-wise predictors \(\{v_{f}^{k}\}_{k=1}^{K}\) based on the available training dataset \(\mathcal{Z}_{f}^{\text{tr}}\). Substituting \(V_{f}\) with \(V_{f}^{(K)}\) defined in (34) into the naive extension of conventional learning in (20) yields the problem  
\[V^{(K),*}(\mathcal{Z}_{f}^{\text{tr}}|\bar{V}^{(K)})=\operatorname* {arg\,min}_{\hat{B}_{f},v_{f}^{k},\ldots,v_{f}^{k}}||X_{f}^{\text{tr}}V_{f}^{(K )}-Y_{f}^{\text{tr}}||_{F}^{2}+\lambda\Big{\|}V_{f}-\bar{V}^{(K)}\Big{\|}_{F}^{2}\] \[V_{f}^{(K)}\!=\!\sum\limits_{k=1}^{K}\!\bar{v}_{f}^{k}\otimes( \tilde{B}_{f}^{k}(\tilde{B}_{f}^{k})^{\dagger})\] subject to \[\tilde{B}_{f}^{\dagger}\hat{B}_{f}=I_{K}, \tag{35}\]  
over the optimization variables \((\hat{B}_{f},\{v_{f}^{k}\}_{k=1}^{K})\). In (35), the hyperparameters \((\lambda,\bar{V}^{(K)})\) are given by the scalar \(\lambda>0\) and by the \(SN\times S\) LSTD-based bias matrix \(\bar{V}^{(K)}\) defined as (cf. (34))  
\[\bar{V}^{(K)}=\sum\limits_{k=1}^{K}\bar{v}^{k}\otimes(\tilde{b}^{k}(\tilde{b} ^{k})^{\dagger}). \tag{36}\]  
Because the Euclidean norm regularization \(\left\|V_{f}-\bar{V}^{(K)}\right\|_{F}^{2}\) in (35) mixes long-term and short-term dependencies due to (34) and (36), we propose the modification of problem (35)  
\[V^{(K),*}(\mathcal{Z}_{f}^{\text{tr}}|\{\tilde{b}^{k},\bar{v}^{k }\}_{k=1}^{K})=\operatorname*{arg\,min}_{\hat{B}_{f},v_{f}^{k},\ldots,\bar{v}^ {k}}\Big{\{}||X_{f}^{\text{tr}}V_{f}^{(K)}-Y_{f}^{\text{tr}}||_{F}^{2}+\lambda _{2}\sum\limits_{k=1}^{K}||v_{f}^{k}-\bar{v}^{k}||^{2}\] \[V_{f}^{(K)}\!=\!\sum\limits_{k=1}^{K}\!\bar{v}_{f}^{k}\otimes( \tilde{B}_{f}^{k}(\tilde{B}_{f}^{k})^{\dagger})\] \[\text{subject to }\tilde{B}_{f}^{\dagger}\hat{B}_{f}=I_{K}, \tag{37}\]  
with hyperparameters \((\lambda_{1},\lambda_{2},\tilde{b}^{1},\ldots,\tilde{b}^{K},\tilde{v}^{1}, \ldots,\tilde{v}^{K})\) given by the scalars \(\lambda_{1},\lambda_{2}>0\), by the \(S\times 1\) long-term bias vectors \(\tilde{b}^{1},\ldots,\tilde{b}^{K}\), and by the \(N\times 1\) short-term bias vectors \(\bar{v}^{1},\ldots,\bar{v}^{K}\). For each feature \(k\), the considered regularization minimizes the Euclidean distance between the short-term prediction vector \(v_{f}^{k}\) and the short-term bias vector \(\bar{v}^{k}\) as in Section 3, while maximizing the alignment between the long-term feature vector \(\tilde{b}_{f}^{k}\) and the long-term bias vector \(\tilde{b}^{k}\) in a manner akin to the kernel alignment method of [56]. To address problem (37), inspired by [57; 58], we propose a sequential approach, in which the pair \((v_{f}^{k},\tilde{b}_{f}^{k})\) consisting of the \(k\)-th predictor \(v_{f}^{k}\) and the \(k\)-th feature vector \(\hat{b}_{f}^{k}\) is optimized in the order \(k=1,2,\ldots,K\). Specifically, at each step \(k\), we consider the problem  
\[\hat{b}_{f}^{k,*},v_{f}^{k,*}=\operatorname*{arg\,min}_{\hat{b}_{f }^{k},v_{f}^{k}} \Big{\{}||X_{f}^{\text{tr  
given the \(k\)-th predictor  
\[(V_{f}^{(K)})^{k}=v_{f}^{k}\otimes(\hat{b}_{f}^{k}(\hat{b}_{f}^{k})^{\dagger}) \tag{40}\]  
and \(k\)-th optimized predictor  
\[(V_{f}^{(K)})^{k,*}=v_{f}^{k,*}\otimes(\hat{b}_{f}^{k,*}(\hat{b}_{f}^{k,*})^{ \dagger}). \tag{41}\]  
Because (38) is a nonconvex problem, we consider alternating least squares (ALS) [59] to obtain the optimal solution \(\{\hat{b}_{f}^{k,*},v_{f}^{k,*}\}\) by iterating between the following steps: (i) for a fixed \(\hat{b}_{f}^{k}\), update \(v_{f}^{k}\) as  
\[v_{f}^{k}\leftarrow \operatorname*{arg\,min}_{ \begin{subarray}{c}v_{f}^{k}\\ (V_{f}^{(K)})^{k}=v_{f}^{k}\otimes(\hat{b}_{f}^{k}(\hat{b}_{f}^{k}))^{\dagger} \end{subarray} } \tag{42}\]  
and (ii) for a fixed \(v_{f}^{k}\), update \(\hat{b}_{f}^{k}\) as  
\[\hat{b}_{f}^{k}\leftarrow\operatorname*{arg\,min}_{ \begin{subarray}{c}b_{f} ^{k}\\ (V_{f}^{(K)})^{k}=v_{f}^{k}\otimes(\hat{b}_{f}^{k}(\hat{b}_{f}^{k})^{\dagger}) \end{subarray}