\subsection{Related Work}
Metric learning (also known as distance learning) has gained significant interest due to its power of effectively learning similarities and dissimilarities among objects. Here, we summarize most relevant contributions from the rich literature on metric learning. \cite{kulis2013metric,bellet2015metric} provide comprehensive summaries of the literature on classical techniques. In this paper, our focus is a specific type of query known as triplet comparisons. There exist methods and efficient algorithms for a variety of feedback such as class labels \citep{weinberger2009distance,davis2007information}, triplet comparisons \citep{schultz2003learning,mason2017learning}, perceptual adjustment queries \citep{xu2024perceptual} and nearest neighbor queries \citep{nadagouda2023active}. A recent study \citep{tatli2024metric} uses triplet comparison queries to perform metric clustering, enabling the discovery of latent subgroups within the population before proceeding to metric learning from triplet comparisons.

\cite{verma2015sample} provide sample complexity of Mahalanobis
distance learning from class labels, which is also known as linear metric learning, where the metric is parametrized by a positive semidefinite matrix. \cite{mason2017learning, ye2019fast} present tight generalization error bounds for Mahalanobis distance metric learning from triplet comparisons. Recently, there has been increased interest in nonlinear metric learning to better fit complex, real-world data sources. Kernelized approaches to the metric learning, similar to the setting considered in this work, are proposed by \cite{martinel2015kernelized, liu2021fast, wang2011metric, chatpatanasiri2010new,kleindessner2017kernel} and many others. More generally, the nonlinear variant has received attention through the study of deep Siamese networks \citep{guo2017learning}.

Recent interest in using deep learning to extract useful representations from  data is followed by triplet network models and its variations \citep{hoffer2015deep}. \cite{kaya2019deep} provide a comprehensive survey on deep metric learning. Despite the empirical success and popularity of deep metric learning techniques on metric learning, theoretical advancements in this area remain sparse. \cite{zhou2024generalization} provides a generalization analysis with deep ReLU networks for metric learning using the hinge loss. Other studies provide generalization guarantees for deep metric learning using neural tangent kernel \citep{liu2021fast} and using Rademacher complexity based analysis \citep{huai2019deep}.

Another line of work focuses on metric learning from pairwise comparisons. Pairwise comparisons can be viewed as a variation of triplet comparisons when it is assumed that there is a reference point $u$ (responder) substituting for leading item $\bx_h$. How to infer preferences from pairwise comparisons is a well studied problem in a diverse set of areas including machine learning, social choice theory, psychology, social sciences and political science (see the work of \cite{furnkranz2010preference} for a comprehensive summary). \cite{xu2020simultaneous} uses a passive algorithm to learn a linear metric and preferences. This can be seen as simultaneously performing metric and preference learning. Later, \cite{canal2022one} extend the results to learning multiple preference points with a shared metric and provide theoretical guarantees for this task. \cite{wang2024metric} analyzes linear metric learning problem with limited pairwise comparisons per user. \cite{chen2024pal} proposes leveraging preference structure to reduce sample complexity of pairwise comparisons.


%\cite{freund2003efficient,burges2005learning,zheng2007regression, xu2020simultaneous} among others propose passive algorithms whereas \cite{jamieson2011active,jamieson2011low,jun2021improved} propose adaptive sample procedures. 