  We consider the problem of ranking a set of items from pairwise comparisons
in the presence of features associated with the items. Recent works have
established that $O(n\log(n))$ samples are needed to rank well when there is no
feature information present. However, this might be sub-optimal in the presence
of associated features. We introduce a new probabilistic preference model
called feature-Bradley-Terry-Luce (f-BTL) model that generalizes the standard
BTL model to incorporate feature information. We present a new least squares
based algorithm called fBTL-LS which we show requires much lesser than
$O(n\log(n))$ pairs to obtain a good ranking -- precisely our new sample
complexity bound is of $O(\alpha\log \alpha)$, where $\alpha$ denotes the
number of `independent items' of the set, in general $\alpha << n$. Our
analysis is novel and makes use of tools from classical graph matching theory
to provide tighter bounds that sheds light on the true complexity of the
ranking problem, capturing the item dependencies in terms of their feature
representations. This was not possible with earlier matrix completion based
tools used for this problem. We also prove an information theoretic lower bound
on the required sample complexity for recovering the underlying ranking, which
essentially shows the tightness of our proposed algorithms. The efficacy of our
proposed algorithms are validated through extensive experimental evaluations on
a variety of synthetic and real world datasets.
