  A popular model of preference in the context of recommendation systems is the
so-called \emph{ideal point} model. In this model, a user is represented as a
vector $\mathbf{u}$ together with a collection of items $\mathbf{x_1}, \ldots,
\mathbf{x_N}$ in a common low-dimensional space. The vector $\mathbf{u}$
represents the user's "ideal point," or the ideal combination of features that
represents a hypothesized most preferred item. The underlying assumption in
this model is that a smaller distance between $\mathbf{u}$ and an item
$\mathbf{x_j}$ indicates a stronger preference for $\mathbf{x_j}$. In the vast
majority of the existing work on learning ideal point models, the underlying
distance has been assumed to be Euclidean. However, this eliminates any
possibility of interactions between features and a user's underlying
preferences. In this paper, we consider the problem of learning an ideal point
representation of a user's preferences when the distance metric is an unknown
Mahalanobis metric. Specifically, we present a novel approach to estimate the
user's ideal point $\mathbf{u}$ and the Mahalanobis metric from paired
comparisons of the form "item $\mathbf{x_i}$ is preferred to item
$\mathbf{x_j}$." This can be viewed as a special case of a more general metric
learning problem where the location of some points are unknown a priori. We
conduct extensive experiments on synthetic and real-world datasets to exhibit
the effectiveness of our algorithm.
