\section{DISCUSSION} \label{sec:conclusion}

When undertaking the task of developing a theoretical understanding of triplet based nonlinear metric learning methods, the first natural setting to consider is kernelized metric learning. To the best of our knowledge, there are no generalization results analyzing the sample complexity for kernelized metric learning via triplet comparisons in the literature prior to our work. The theoretical foundations for metric learning via triplet queries are currently limited to linear settings, e.g., \cite{mason2017learning}, which provide generalization results for the linear setting when the set of items being queried is fixed and the number of items $n\gg d$ (See Appendix \ref{appendix:discussion} for further explanation). Therefore, our work fills an important gap in the literature. %. It is an important step towards understanding nonlinear methods in metric learning.

We provide a theoretical analysis for the kernelized metric learning problem. We provide novel generalization and sample complexity bounds. Developing an understanding of other nonlinear metric learning approaches, especially neural networks based approaches would be interesting for future research directions. That said, kernelized approaches are preferred in areas where interpretability and explainability are crucial, especially when they also perform nearly as well as other methods \citep{radhakrishnan2023transfer}. Therefore, understanding kernelized settings is also of value beyond theoretical pursuit towards understanding a broader set of nonlinear approaches. 

%Additionally, building on existing work in simultaneous metric and preference learning, we plan to expand our analysis to incorporate user preferences. This introduces new challenges, such as generalization to previously unseen users.
