%\documentclass{uai2025} % for initial submission
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\title{Metric Learning in an RKHS}

% The standard author block has changed for UAI 2025 to provide
% more space for long author lists and allow for complex affiliations
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% All author information is authomatically removed by the class for the
% anonymous submission version of your paper, so you can already add your
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% Add authors
\author[1]{\href{mailto:<gtatli@wisc.edu>?Subject=Your UAI 2025 paper}{Gokcan~Tatli}{}}
\author[1]{Yi~Chen}
\author[2]{Blake~Mason}
\author[1]{Robert~Nowak}
\author[1]{Ramya~Korlakai~Vinayak}
% Add affiliations after the authors
\affil[1]{%
    Dept. of Electrical and Computer Engineering\\
    University of Wisconsin-Madison\\
    Madison, Wisconsin, USA
}
\affil[2]{%
    Amazon.com\\
    USA
}

  \input{preamble}
\input{macros}
  \begin{document}
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\begin{abstract}
  This paper investigates metric learning in a Reproducing Kernel Hilbert Space (RKHS) based on a set of random triplet comparisons in the form of ``\textit{Do you think item h is more similar to item i or item j?}" indicating similarity and differences between various items. The goal is to learn a metric in the RKHS that reflects the comparisons. Nonlinear metric learning using kernel methods and neural networks has shown great empirical promise. While previous works have addressed certain aspects of this problem, there is little or no theoretical understanding of such methods. The exception is the special (linear) case in which the RKHS is the standard $d$-dimensional Euclidean space; there is a comprehensive theory for metric learning in the $d$-dimensional Euclidean space. This paper develops a general RKHS framework for metric learning and provides novel generalization guarantees and sample complexity bounds. We validate our findings through a set of simulations and experiments on real datasets. Our code is publicly available at \texttt{\url{https://github.com/RamyaLab/metric-learning-RKHS}}.
\end{abstract}

\input{sections/Introduction}
\input{sections/Related Work}
\input{sections/ProblemSetting}
\input{sections/KernelizedML}
\input{sections/Theoretical Guarantees}
\input{sections/Practical Implementation}
\input{sections/KPCA}
\input{sections/subsections_Learning in RKHS/Calculatingnorm}
\input{sections/subsections_Learning in RKHS/Relatingnorms}
\input{sections/LLowDimKernel}
\input{sections/Experiments}
\input{sections/Conclusion}
\section{Acknowledgements}
This work was partially supported by NSF grants NCS-FO 2219903 and NSF CAREER Award CCF 2238876.
\bibliography{kernelmetric.bib}

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\title{Metric Learning in an RKHS\\(Supplementary Material)}
\maketitle

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%\input{sections/Supplementary/Outline}

\input{sections/Supplementary/Broader Impacts}
\input{sections/Supplementary/Proofs}
\input{sections/Supplementary/Discussion}
\input{sections/Supplementary/Additional Details}

\end{document}
