## 3 Proposed Algorithm
### Class-Wise Classifier Design Capable of Continual Learning  
Figure 1 shows the overview of CAEAC. The architecture of CAEAC is inspired by ASC [8]. As shown in Figure 2, ASC is an ASOINN-based supervised classifier incorporating \(k\)-means and two node clearance mechanisms after class-wise unsupervised self-organizing processes. The main difference between CAEAC and ASC is that CAEAC does not require \(k\)-means and node clearance mechanisms because CAEA has superior clustering performance to ASOINN.  
In CAEAC, a training dataset is divided into multiple subsets based on their class labels. The number of the subsets is the same as the number of classes. Each subset is used to generate a classifier (i.e., nodes and edges) through a self-organization process by CAEA. Since CAEA is capable of continual learning, each classifier can be continually updated.  
\begin{table}
\begin{tabular}{l l} \hline \hline
**Notation** & **Description** \\ \hline \(k_{1},k_{2}\) & Indexes of the 1 st and 2 nd winner nodes \\ \(\mathbf{y}_{k_{1}},\mathbf{y}_{k_{2}}\) & The 1 st and 2 nd winner nodes \\ \(V_{k_{1}},\mathbf{y}_{k_{2}}\) & Similarities between a data point \(\mathbf{x}_{n}\) and winner nodes (\(\mathbf{y}_{k_{1}}\) and \(\mathbf{y}_{k_{2}}\)) \\ \(V_{\text{threshold}}\) & Similarity threshold (a vigilance parameter) \\ \(\mathcal{N}_{k_{1}}\) & A set of neighbor nodes of node \(\mathbf{y}_{k_{1}}\) \\ \(\alpha_{k_{1}}\) & The number of data points that have accumulated by the node \(\mathbf{y}_{k_{1}}\) \\ \(\lambda\) & Predefined interval for computing \(\sigma\) and deleting an isolated node \\ \(e_{(k_{1},k_{2})}\) & Edge connection between nodes \(\mathbf{y}_{k_{1}}\) and \(\mathbf{y}_{k_{2}}\) \\ \(a_{(k_{1},k_{2})}\) & Age of edge \(e_{(k_{1},k_{2})}\) \\ \(a_{\text{max}}\) & Predefined threshold of an age of edge \\ \hline \hline \end{tabular}
\end{table}
Table 1: _Cont._  
Figure 1: Overview of CAEAC.  
Figure 2: Overview of ASC. Reprinted/adapted with permission from Ref. [8].  
Moreover, when a training dataset belongs to a new class, a self-organizing space of CAEA is newly defined. Thus, it is possible to learn new features/information without destroying existing one. When classifying an unknown data point, classifiers (one classifier for each class) are installed in the same space, and the label information of the nearest neighbor node of the unknown data point is output as a classification result. The learning procedure of CAEAC is summarized in Algorithm 1.  
```
Input: the training data points: \(\mathcal{X}=\{\mathbf{x}_{1},\mathbf{x}_{2},\ldots,\mathbf{x}_{L}\}(\mathbf{x }_{l}\in\mathbb{R}^{d})\), the predefined interval for computing \(\sigma\) and deleting an isolated node: \(\lambda\), and the predefined threshold of an age of edge: \(a_{\max}\). Output: the CAEA models.
1 functionLearningCAEAC(\(\mathcal{X},\lambda\), \(a_{\max}\))
2 forall\(c\in 1,2,\ldots,C\)do
3\(\mathrm{model}_{c}=\mathrm{LearningCAE}(\mathcal{X}_{c},\lambda\), \(a_{\max}\)).
4 returnCAEA models.
```  
**Algorithm 1**Learning Algorithm of CAEAC  
In the following subsections, the mathematical backgrounds of the CIM and CAEA are explained in detail.