# U-Vectors: Generating Clusterable Speaker Embedding from Unlabeled Data
## 3 Methodology
### Speech Segments  
To generate the pairwise constraints, it is considered that a speech segment belongs to an individual. Moreover, if it is possible to extract accurate pairwise constraints, a DL framework can be trained using those constraints. To generate such pairwise constraints, speech segmentation procedures are required.  
\begin{table}
\begin{tabular}{c p{227.6 pt}} \hline \hline
**Notation** & **Description** \\ \hline \(\mathcal{S}\) & A set of audio segments. Audio segments are fragments of a continuous audio stream. We assume that most audio segments contain speech of a single individual. \\ \hline \(\mathcal{X}\) & A single audio segment, \(\mathcal{X}\in\mathcal{S}\). \\ \hline \(x_{i}\) & An audio frame, generated by taking shorter frames from an audio segment, \(x_{i}\in\mathcal{X}_{k}\). Audio frames are used to train DL architectures. \\ \hline \(\mathcal{M}\) & Denotes the number of possible audio frames in an audio segment, \(x_{1\leq i\leq\mathcal{M}}\in\mathcal{X}_{k}\). Theoretically, \(\mathcal{M}\times|x_{i}|=|\mathcal{X}_{k}|\). \\ \hline \(\mathcal{C}\) & A set of clusters. These clusters are formed using hypothetical pairwise constraints. As cluster linkages are constructed based on the speech segment relations, it can be considered that \(|\mathcal{S}|=|\mathcal{C}|\). \\ \hline \(c_{i}\) & Denotes a subset of the entire cluster, \(c_{i}\subseteq\mathcal{C}\). Here, \(c_{i}\) represents a cluster constructed using the inter-relationship of speech frames, belonging to a specific speech segment \(\mathcal{X}_{i}\). \\ \hline \(\mathcal{N}\) & The actual number of individuals in \(\mathcal{S}\), considering the ground truth. For this specific problem, the value of \(\mathcal{N}\) is unknown. \\ \hline \(\alpha\) & The distance hyperparameter used for AutoEmbedder [21] architecture. For other architectures, \(\alpha\) may indicate a connectivity state for any two cluster nodes. \\ \hline \hline \end{tabular}
\end{table}
Table 1: The mathematical notations used in the paper are summarized.  
Speech can be easily segmented using various techniques. Methods such as VAD [34] and word segmentation [35] can be indeed adopted to define such speech segments, containing the voice of a single individual. It is also feasible to assume that a single individual mostly speaks more than one word in a conversation. Hence, it is also possible to queue multiple speech segments and hypothesize that they come from a single individual. However, increasing the queue or size of a speech segment also increases the probability of impurity of a speech segment (discussed in Section 3.5). By impurity in a speech segment, it is referred that a speech segment contains more than one speaker. Impure data can often trick the DL frameworks from finding actual relationships among clusters. Hence, to minimize the impurity risk, we study speech segments with a length of one second. After the successful extraction of speech segments, the pairwise constraints are to be constructed. Although the overall framework is dependent on proper speech segmentation techniques, we avoid implementing such segmentation methods. Instead, we provide a detailed evaluation of embedding accuracy based on various levels of cluster impurities.