## 2 Principle of the Proposed Scheme
### Principle of CS M-ary SVM  
As we already know, the distribution of PS-64 QAM constellation points is uneven. For the SVM algorithm, an uneven distribution constellation will impel the hyperplane to move to the lower probability constellation points as close as possible. Therefore, a large number of outer constellation points will be mistakenly considered as inner constellation points. In order to solve the problem, we proposed the CS M-ary SVM NLE scheme. The process of the CS M-ary SVM NLE scheme is shown in Figure 5. Firstly, the constellation of PS-64 QAM is divided into two subsets by a SVM. In this step, the training data (60% of the data set) are used to obtain a hyperplane, and the testing data (40% of the data set) are divided into two subsets according to the hyperplane. Then the inner and outer sets are detected by the M-ary SVM scheme, respectively. Figure 6 shows the Q factor as a function of launched optical power for 120 Gb/s PS-64 QAM with different constellation segment methods. The Q factor can be calculated by BER and the relationship between them can be expressed as \(Q=20 log_{10}\Big{[}\sqrt{2}erfc^{-1}(2 BER)\Big{]}\)[23]. For PS-64 QAM, the constellation points have nine different amplitudes and the points with the same amplitude have the same probability, which can be calculated by the Formula (1). The amplitude and probability distributions of the constellation points are shown in Table 1. In the step of the constellation segment, the training data are labeled as \"0\" or \"1\" to identify the points belonging to the inner set or the outer set. We label the training data according to the amplitudes of the corresponding standard constellation points which have not been transmitted in the optical communication system. The hyperplane obtained by the training data is used to divide the testing data into two subsets. According to the amplitudes of the standard constellation points, there are eight  
constellation points is 0.549824, which is the closest to 0.5 compared with the other seven constellation segment methods. Therefore, in this scheme, we pick the situation in which the number of inner constellation points is 12 and the number of outer constellation points is 52 as the constellation segment method.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**The Number of** & **The Amplitude of Each** & **The Probability of Each** \\
**Constellation Points** & **Constellation Point** & **Constellation Point** \\ \hline
4 & \(\sqrt{2}\) & 0.058564 \\
8 & \(\sqrt{10}\) & 0.039446 \\
4 & \(\sqrt{18}\) & 0.026569 \\
8 & \(\sqrt{26}\) & 0.017666 \\
8 & \(\sqrt{34}\) & 0.011899 \\
12 & \(\sqrt{50}\) & 0.005324 \\
8 & \(\sqrt{58}\) & 0.003586 \\
8 & \(\sqrt{74}\) & 0.001606 \\
4 & \(\sqrt{98}\) & 0.000484 \\ \hline \hline \end{tabular}
\end{table}
Table 1: The amplitude and probability distributions of the standard PS-64 QAM constellation (\(\
u=0.05\)).  
Figure 5: The process of the CS M-ary SVM NLE scheme.  
Figure 6: \(Q\) factor versus launched optical power for 120 Gb/s PS-64 QAM with different constellation segment methods.  
For the constellation segment method we choose in this paper, the number of inner constellation points is 12 and the number of outer constellation points is 52. For the inner constellation points, the second bits of all constellation points are 1, while the fifth bits of all constellation points are 1 as well. Therefore, only 4 SVMs (Inner-SVM1, Inner-SVM3, Inner-SVM4 and Inner-SVM6) are required to detect the symbol of the inner subset whose classification strategy is shown in Figure 7a. However, there are six SVMs needed for the outer set detection, whose classification strategy is shown in Figure 7b. Finally, we obtain the mapping symbol and calculate the BER.