  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.
