  This paper investigates simultaneous preference and metric learning from a
crowd of respondents. A set of items represented by $d$-dimensional feature
vectors and paired comparisons of the form ``item $i$ is preferable to item
$j$'' made by each user is given. Our model jointly learns a distance metric
that characterizes the crowd's general measure of item similarities along with
a latent ideal point for each user reflecting their individual preferences.
This model has the flexibility to capture individual preferences, while
enjoying a metric learning sample cost that is amortized over the crowd. We
first study this problem in a noiseless, continuous response setting (i.e.,
responses equal to differences of item distances) to understand the fundamental
limits of learning. Next, we establish prediction error guarantees for noisy,
binary measurements such as may be collected from human respondents, and show
how the sample complexity improves when the underlying metric is low-rank.
Finally, we establish recovery guarantees under assumptions on the response
distribution. We demonstrate the performance of our model on both simulated
data and on a dataset of color preference judgements across a large number of
users.
