## 6 Conclusions  
Although many studies have been conducted on animal identification, the performance improvement has been limited to the application of the existing face identification methodology owing to the low normality of dog faces and the lack of related data. This paper proposed to overcome these limitations with a novel loss function and learning method that utilize a wide vector space and thereby improves the performance of the dog face identification model. The triplet loss used in the existing face identification model was trained based on cosine similarity using the L2-norm layer. This paper proposes the training of a model without an L2-norm layer using vector length loss, which is a novel loss function for the length of the embedding vector, and its two-stage learning method. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Learning**} & \multicolumn{3}{c}{**Rank 1**} & \multicolumn{3}{c}{**Rank 5**} \\ \cline{2-7}  & **Mean** & **Maximum** & **Minimum** & **Mean** & **Maximum** & **Minimum** \\ \hline SOTA & 10.96\% & 14.79\% & 6.67\% & 32.58\% & 38.54\% & 26.25\% \\ \hline Triplet & 37.52\% & 45.21\% & 31.04\% & 65.84\% & 72.50\% & 59.58\% \\ \hline ArcFace & 38.41\% & 44.38\% & 32.92\% & 65.92\% & 70.83\% & 60.83\% \\ \hline Triplet + VL & 39.74\% & 47.08\% & 32.92\% & 68.80\% & 73.96\% & 63.12\% \\ \hline ArcFace + VL & 34.92\% & 41.67\% & 28.96\% & 67.57\% & 73.12\% & 61.25\% \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Results of mean, maximum, and minimum accuracy of rank 1 and rank 5 for the learning methods in face identification. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Learning Methods** & **Threshold** & **Accuracy** & **Threshold** & **Accuracy** \\ \hline SOTA & \(\alpha\) = 0.3 & 76.0\% & _best_ = 0.38 & 76.9\% \\ \hline Triplet & \(\alpha\) = 0.3 & 55.9\% & _best_ = 1.29 & 87.0\% \\ \hline ArcFace & \(\alpha\) = 0.3 & 55.2\% & _best_ = 1.31 & 86.4\% \\ \hline Triplet + VL & \(\alpha\) = 0.3 & 54.3\% & _best_ = 2.49 & 88.4\% \\ \hline ArcFace + VL & \(\alpha\) = 0.3 & 60.2\% & _best_ = 1.49 & 88.8\% \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Results of the mean of the optimum accuracy and the threshold for the learning methods in face verification. The embedding vector was distributed over the surface of the sphere into a wider vector space, which improved the performance of the dog face identification model by 4%. The distribution of embedding vectors is limited compared to the infinite vector space, but this paper proposes the novel loss function and learning method to utilize the vector space wider than the surface of the sphere. The proposed methodology improves the model's performance compared to the existing methodology for the same model. As a result, this paper overcomes the limitation of triplet loss in face identification and improves the performance of the dog face identification model, which is expected to help return lost dogs. Furthermore, the model's performance will be further improved by using the proposed loss function and its learning method for other models and datasets using the existing triplet loss and L2-norm layer. Conceptualization, J.R.; methodology, B.Y. ; software, B.Y. and H.S. ; validation, B.Y. and H.S. ; formal analysis, B.Y. ; investigation, B.Y. ; resources, B.Y. ; data curation, B.Y. ; writing--original draft preparation, B.Y. ; writing--review and editing, B.Y. and H.S. ; visualization, H.S. ; supervision, J.R.; project administration, J.R.; funding acquisition, J.R. All authors have read and agreed to the published version of the manuscript. This work was carried out with the support of \"Cooperative Research Program of Center for Companion Animal Research (Project No. PJ0139862020)\" Rural Development Administration, Korea. Not applicable. Publicly available datasets were analyzed in this study.