We study metric learning from preference comparisons under the ideal point
model, in which a user prefers an item over another if it is closer to their
latent ideal item. These items are embedded into $\mathbb{R}^d$ equipped with
an unknown Mahalanobis distance shared across users. While recent work shows
that it is possible to simultaneously recover the metric and ideal items given
$\mathcal{O}(d)$ pairwise comparisons per user, in practice we often have a
limited budget of $o(d)$ comparisons. We study whether the metric can still be
recovered, even though it is known that learning individual ideal items is now
no longer possible. We show that in general, $o(d)$ comparisons reveals no
information about the metric, even with infinitely many users. However, when
comparisons are made over items that exhibit low-dimensional structure, each
user can contribute to learning the metric restricted to a low-dimensional
subspace so that the metric can be jointly identified. We present a
divide-and-conquer approach that achieves this, and provide theoretical
recovery guarantees and empirical validation.

度量学习、偏好比较、理想点模型、马氏距离、低维结构

有限配对偏好比较的度量学习

Metric Learning from Limited Pairwise Preference Comparisons

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.

本文介绍了一种新的方法，该方法针对推荐系统中的偏好建立了一种 Mahalanobis 度量，进而学习理解用户对物品的理想点。