We explore the metric and preference learning problem in Hilbert spaces. We obtain a novel representer theorem for the simultaneous task of metric and preference learning. Our key observation is that the representer theorem can be formulated with respect to the norm induced by the inner product inherent in the problem structure. Additionally, we demonstrate how our framework can be applied to the task of metric learning from triplet comparisons and show that it leads to a simple and self-contained representer theorem for this task. In the case of Reproducing Kernel Hilbert Spaces (RKHS), we demonstrate that the solution to the learning problem can be expressed using kernel terms, akin to classical representer theorems.

翻译：我们探讨希尔伯特空间中的度量与偏好学习问题。针对度量与偏好学习的联合任务,我们提出了一种新颖的表示者定理。关键发现是,该表示者定理可以基于问题结构内在内积所诱导的范数进行表述。此外,我们展示了该框架如何应用于三元组比较的度量学习任务,并证明其能为此任务导出简洁自洽的表示者定理。对于再生核希尔伯特空间(RKHS)情形,我们证明学习问题的解可借助核项表达,这与经典表示者定理的形式一致。