## 3 Models and Algorithms
### Selecting the Reference Channels  
Let us first describe the selection of the set of RChs. It is worth noting that the \(\sigma^{\text{I}}\) and \(\sigma^{\text{Q}}\) features provide insights into the dispersion of the symbols around the mean of each CP. Therefore, we focus on analyzing these features to determine the smallest set of RChs that best represents the behavior of the complete set of channels in the spectrum.  
The algorithm starts by retrieving a set of signal samples from the data repository, as well as the BER measured when those samples were obtained (line 1 in Algorithm 1). Next, features \(Y\) are computed using GMM fitting (line 2), which are subsequently filtered to select \(\sigma^{I}\) and \(\sigma^{\text{Q}}\) for the selected CPs (line 3).  
The proposed approach relies on fitting piecewise linear functions representing the behavior of the \(\sigma\) features. The cut-points of the function are then candidates to become reference channels. The pseudocode is presented in Algorithm 1, which receives the order of MF, the set of selected CPs, and the largest admissible distance for the considered network (in km). It is clear that having a large number of reference channels will provide better accuracy and simplify the subsequent feature composition, However, that will increase the complexity of the DNN model pre-training process, Therefore, we target selecting just the smallest number of reference channels that provides good accuracy without adding excessive complexity to the features composition. In consequence, the algorithm analyzes the results of selecting a specific number of reference channels, which is increased until the desired accuracy is achieved (lines 4-11), or a maximum number of reference channels is exceeded (line 12). At every iteration, the algorithm fits each \(\sigma\) feature with the desired linear segments to obtain a list of cut-points, which are collected into a single set of candidate reference channels (lines 6-8). As a fitting function in line 7, we select a piecewise linear approach, due to its low complexity, easiness of interpretability, and ability to find the cut-points that allow the highest accuracy with simple linear segments, which extols the high relevance of the selected cut-points.  
When all the features are processed, the set of candidate channels contains \(i\) clusters with possible reference points, so the next step (line 9) finds the centroids for each cluster, which are the candidate reference channels for this iteration. Then, the pre-FEC BER is estimated over these centroids (line 10) and compared with the real BER. In case the relative error is smaller than the target threshold, the candidates become the reference channels to be used and are returned (line 11). Otherwise, the process continues until the maximum admissible number of reference channels is reached (line 12).  
\begin{table}
\begin{tabular}{l l} \hline \hline
**General** & \\ \hline \(X\) & Set of symbols, index \(x\). \\ \(\gamma^{i}\) & Features of CP \\ \(m\) & Order of modulation format and number of CPs \\ _selCP_ & Set of selected CPs to be propagated by the OCATA DNN models \\ \hline \hline
**NLI noise mitigation** & \\ \hline \(A\) & The whole coordinate plane, which is divided into small squares \(a\). \\ \(A^{i}\) & Detection area for CP \(i\). \\ \(k\) & Number of squares areas \(a\), each of size \(\delta\times\delta\) \\ \(\mathcal{D}\) & Square matrix of size sqrt(\(k\)) \(\times\) sqrt(\(k\)) for symbol decoding. \\ \(\phi_{A}(i)\) & Probability that a symbol from CP \(i\) is received inside the detection area \(A^{i}\). \\ \(\phi_{out}\) & Complementary probability of \(\phi_{A}(i)\). \\ \(\varphi_{u}(i)\) & Probability of a symbol from CP \(i\) is received inside the small square \(a\in A^{i}\). \\ \hline \hline \end{tabular}
\end{table}
Table 1: Notation.