## 2 Principle  
### Concept of Support Vector Machine  
SVM is a powerful supervised machine learning algorithm which can be applied for amplitude and phase noise characterization [22]. SVM-based soft decision for QAM and APSK constellation has been proposed in optical communication system, and shown great improvement especially when there is strong nonlinearity. Because traditional hard decision is based on the Euclidean distance of separate symbol, it only works well when the noise distribution follows independent identical distribution for all levels. SVM, on the other hand, can provide accurate classification boundary according to the characteristics of noise distribution through training data set. Therefore, SVM-based soft decision is superior to Euclidean distance-based hard decision.  
In general, the in-phase and quadrature (I/Q) components are concatenated as 2-D feature vectors. If the training data set is not linearly separable, the kernel function can be used to map the data from the lower-dimensional feature space to a higher-dimensional feature space [12], so that the data can be linearly separated. According to the statistical learning theory, the main function of SVM is to find an optimal hyperplane that ensures the minimum distance from the optimal hyperplane to the nearest points of each group is maximum, namely the maximizing margin rule [14; 23]. Given the function of the hyperplane,  
\[\mathbf{w}^{T}\cdot\mathbf{x}-b=0, \tag{1}\]  
where \(\mathbf{x}\) is the feature vector, and \(\mathbf{w}\) and \(b\) denote the weight and bias. Equation (2) describes the max-margin problem [23]:  
\[\begin{array}{ll}\underset{\{\mathbf{w},b\}}{\min}&\frac{1}{2}\|\mathbf{w}\| +C\sum\limits_{i=1}^{n}\xi_{i}\\ \text{s. t.}&\mathbf{y}_{i}(\mathbf{w}^{T}\cdot\mathbf{x}_{i}-b)\geq 1-\xi_{i} \end{array} \tag{2}\]  
where \(2/\|\mathbf{w}\|\) is the margin, and \(\mathbf{y}i\in\{\pm 1\}\) denoting the binary class of \(\mathbf{x}_{i}\). The penalty parameter \(C\) and the non-negative slack variable \(\xi_{i}\) are introduced to provide a soft margin [23]. \(C\) is also related to the box constraint, which controls the penalty imposed on margin-violating observations [24]. A small box constraint provides a more flexible model that can prevent overfitting. The points closest to the hyperplane are named support vectors. In other words, the trained optimal hyperplane only depends on a few support vectors.  
As SVM is originally proposed for binary classification, for data set with more than two classes, additional multi-class SVM strategy is required to decompose the multi-class problem into several binary classifiers. Common multi-class strategies include one-versus-one (OVO), one-versus-rest (OVR), and directed acyclic graph (DAG) [13]. The number of required classifiers for different multi-class strategies is displayed in Table 1. For \(m\)-QAM constellation, bit-based binary classifiers [13; 14; 15] can achieve excellent performance with only \(\log_{2}m\) classifiers. The bit-based binary SVM is effective for \(m\)-QAM signal with an even \(\log_{2}m\), because the constellation points are evenly distributed in a lattice and the bit-to-symbol mapping follows gray coding, the adjacent symbols have one single bit differing. Intuitive bit-based classification hyperplanes can be easily obtained.