## 3 Proposed Method
### Decision Tree-Based Segment Clustering  
We then perform decision tree-based clustering from fixed-length vectors to find subword units consisting of acoustically similar vectors. Decision tree-based clustering, in HMM-based ASR, is generally used for HMM state clustering and testing methods to address the data sparsity issue and address unfamiliar contexts [4]. In addition, a previous study [11] used this clustering algorithm to derive phoneme-like units from context-dependent grapheme units. Similarly, we use decision tree-based clustering to derive new subword units from the fixed-length vectors to obtain phoneme-level segments. The proposed method, as shown in Figure 3a, works in three sub-steps: obtaining statistics, building questions, and constructing a decision tree. The clustering algorithm was implemented by modifying a clustering method provided by the Kaldi toolkit [29] for our purposes (e.g., obtaining statistics on the fixed-length vectors instead of the feature vectors). The decision tree was implemented via top-down greedy splitting. Each question for a split is automatically generated by top-down binary clustering of the data to ensure the highest likelihood under the node in the corresponding branches. To build the decision tree, we first collect cumulative statistics for each CD phoneme unit into a statistical table as shown in Figure 3a. Here, the CD phoneme unit uses a tri-phone that considers both, the left and the right neighboring phonemes. Our method generates a statistical table using the segment-level fixed-length vectors, unlike the previous method [11] using frame-level feature vectors. The statistical table consists of the number of samples (_count_), the sum of the fixed-length vectors (\(x\)), and the sum of the squares of the fixed-length vectors (\(x^{2}\)). The statistical table is then used to estimate the weight, mean, and covariance for calculating the log-likelihood of a single Gaussian value at each node. The next steps of building questions and building a decision tree are performed in the same way as the decision tree-based clustering method using the Kaldi toolkit [29]. In the step involving building questions, we generate questions by using phoneme groups and the original phoneme set, as shown  
Figure 3: Block diagram and a simple example of decision tree-based clustering. (**a**) Block diagram of the decision tree-based clustering; (**b**) statistical table obtained at point A; (**c**) questions obtained at point B; (**d**) decision tree obtained at point C.  
in Figure 3b. Here, for simplicity, we assume that only four phonemes, \(a\), \(b\), \(c\), and \(d\), exist. To obtain the phoneme groups, we first build a decision tree with a single root node and put all the phonemes in the root node (Q0). Then, we select a node with the highest log-likelihood improvement in the decision tree and find the question with the highest log-likelihood difference when dividing the node into the left branch and the right branch. In Figure 3c, the phoneme group tree shows the maximum log-likelihood difference when the node Q0 {\(a,b,c,d\)} is split into the left branch \(Q1\){\(a,b\)} and the right branch \(Q2\){\(c,d\)}. We repeat this process until the total number of leaf nodes is the number of phonemes. As a result, we generate nine questions (Q0-Q8) consisting of five phoneme groups (Q0-Q4) and four original phonemes (Q5-Q8). Equations (1) and (2) show the mean and covariance of each node, where \(N_{p}\) is the total _count_ of data belonging to node \(p\). Equation (3) represents a single Gaussian value of the \(D\)-dimensional fixed-length vectors belonging to node \(p\). Finally, Equations (4) and (5) show the log-likelihood \(L_{p}\) for node \(p\) and the log-likelihood difference \(\Delta L\).