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A popular model of preference in the context of recommendation systems is theso-called \emph{ideal point} model. In this model, a user is represented as avector $\mathbf{u}$ 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 thatrepresents a hypothesized most preferred item. The underlying assumption inthis 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 vastmajority of the existing work on learning ideal point models, the underlyingdistance has been assumed to be Euclidean. However, this eliminates anypossibility of interactions between features and a user's underlyingpreferences. In this paper, we consider the problem of learning an ideal pointrepresentation of a user's preferences when the distance metric is an unknownMahalanobis metric. Specifically, we present a novel approach to estimate theuser's ideal point $\mathbf{u}$ and the Mahalanobis metric from pairedcomparisons 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 metriclearning problem where the location of some points are unknown a priori. Weconduct extensive experiments on synthetic and real-world datasets to exhibitthe effectiveness of our algorithm.

Simultaneous Preference and Metric Learning from Paired Comparisons