## 4 Multi-Antenna Frequency-Selective Channels
### Meta-Learning for LSTD-Based Prediction  
Plugging (37) into the naive extension of (22), we can formulate the meta-learning problem for LSTD-based prediction as  
\[\min_{\{\bar{b}^{k},\bar{\sigma}^{k}\}_{k=1}^{K}}\sum_{f=1}^{F}\Big{\|}X_{f}^{ \text{te}}V^{(K),*}(Z_{f}^{\text{tr}}|\{\bar{b}^{k},\bar{\sigma}^{k}\}_{k=1}^{K} )-Y_{f}^{\text{te}}\Big{\|}_{F}^{2}. \tag{49}\]  
Similar to the sequential approach (38) described in Section 4.4, we propose a hierarchical sequential approach for meta-learning by using (38) in the order \(k=1,\ldots,K\), obtaining the problem  
\[\bar{b}^{k,\text{meta}},\sigma^{k,\text{meta}}=\underset{\bar{b}^{k},\bar{ \sigma}^{k}}{\arg\min}\ \Big{\{}\sum_{f=1}^{F}\Big{\|}X_{f}^{\text{te}}(V_{f}^{(K)})^{k,*}-(Y_{f}^{ \text{te}})^{k}\Big{\|}_{F}^{2}\Big{\}}, \tag{50}\]  
with the residual target matrix \((Y_{f}^{\text{te}})^{k}\) defined as (cf. (39))  
\[(Y_{f}^{\text{te}})^{k}= \begin{cases}Y_{f}^{\text{te}},\text{ for }k=1,\\ Y_{f}^{\text{te}}-\sum_{k^{\prime}=1}^{k-1}X_{f}^{\text{te}}(V_{f}^{(K)})^{k^{ \prime},*},\text{ for }k>1.\end{cases}  \tag{51}\]  
The bilevel non-convex optimization problem (50) is addressed through gradient-based updates with gradients computed via equilibrium propagation (EP) [60; 61]. EP uses finite differentiation to approximate the gradient of the bilevel optimization (50), where the difference is computed between two gradients obtained at two stationary points \((\hat{b}_{f}^{k,*},v_{f}^{k,*})\) and \((\hat{b}_{f}^{k,*},v_{f}^{k,*})\) for the original problem (38) and modified version of (38) that considers additional prediction loss for the test set \(\mathcal{Z}_{f}^{\text{te}}\). Specifically, EP leverages the asymptotic equality [60]  
\[\
abla_{\bar{b}^{k}}\sum_{f=1}^{F}\Big{\|}X_{f}^{\text{te}}(V_{f}^{(K)})^{k,*} -(Y_{f}^{\text{te}})^{k}\Big{\|}_{F}^{2}=\lim_{\alpha\to 0}\frac{2\lambda_{1}}{ \alpha}\sum_{f=1}^{F}\Big{(}\hat{b}_{f}^{k,*}(\hat{b}_{f}^{k,*})^{\dagger}- \hat{b}_{f}^{k,\alpha}(\hat{b}_{f}^{k,\alpha})^{\dagger}\Big{)}\bar{b}^{k} \tag{52}\]  
and  
\[\
abla_{\bar{b}^{k}}\sum_{f=1}^{F}\Big{\|}X_{f}^{\text{te}}(V_{f}^{(K)})^{k,*} -(Y_{f}^{\text{te}})^{k}\Big{\|}_{F}^{2}=\lim_{\alpha\to 0}\frac{2\lambda_{2}}{ \alpha}\sum_{f=1}^{F}(v_{f}^{k,*}-v_{f}^{k,\alpha}), \tag{53}\]  
with additional real-valued hyperparameter \(\alpha\in\mathbb{R}\), which is generally chosen to be a non-zero small value [60; 61]. In (52) and (53), vectors \(\hat{b}_{f}^{k,\alpha}\) and \(v_{f}^{k,\alpha}\) are defined as (cf. (38))  
\[\hat{b}_{f}^{k,\alpha},v_{f}^{k,\alpha}=\underset{\bar{b}^{k},\bar{ \sigma}^{k}}{\arg\min}\Big{\{}||X_{f}^{\text{tr}}(V_{f}^{(K)})^{k  
After meta-learning, as in Section 3.3, based on the optimized \(\bar{b}^{\text{1,meta}},\ldots,\bar{b}^{K,\text{meta}}\) and \(\bar{v}^{\text{1,meta}},\ldots,\bar{v}^{\text{K,meta}}\), LSTD-based channel predictor for a new frame \(f^{\text{new}}\) can be obtained via (37) as  
\[V_{f^{\text{new}}}^{(K),*}=V^{(K),*}(\mathcal{Z}_{f^{\text{new}}}^{\text{tr}}| \{\bar{b}^{\text{k,meta}},\phi^{\text{k,meta}}\}_{k=1}^{K}), \tag{55}\]  
which can be solved in a sequential way, as in (38). The computational complexity order of the considered schemes is summarized in Tables 1 and 2. At deployment time, as seen in Table 1, all schemes require the same computational complexity of conventional learning. In contrast, in the offline meta-learning or transfer learning phase, the computational overhead depends on the dimension of the channel vector \(S=N_{R}N_{T}W\). LSTD-based schemes can reduce the computational overhead as compared to naive solutions when the channel vector is large, i.e., \(S\gg 1\), and the rank \(K\) is sufficiently small. This is quantified in Table 2, where \(I_{\text{ALS}}\) is the number of iterations for ALS and \(I_{\text{EP}}\) is the number of iterations for EP.