  We consider the following problem, which is useful in applications such as
joint image and shape alignment. The goal is to recover $n$ discrete variables
$g_i \in \{0, \ldots, k-1\}$ (up to some global offset) given noisy
observations of a set of their pairwise differences $\{(g_i - g_j) \bmod k\}$;
specifically, with probability $\frac{1}{k}+\delta$ for some $\delta > 0$ one
obtains the correct answer, and with the remaining probability one obtains a
uniformly random incorrect answer. We consider a learning-based formulation
where one can perform a query to observe a pairwise difference, and the goal is
to perform as few queries as possible while obtaining the exact joint
alignment. We provide an easy-to-implement, time efficient algorithm that
performs $O\big(\frac{n \lg n}{k \delta^2}\big)$ queries, and recovers the
joint alignment with high probability. We also show that our algorithm is
optimal by proving a general lower bound that holds for all non-adaptive
algorithms. Our work improves significantly recent work by Chen and Cand\'{e}s
\cite{chen2016projected}, who view the problem as a constrained principal
components analysis problem that can be solved using the power method.
Specifically, our approach is simpler both in the algorithm and the analysis,
and provides additional insights into the problem structure.
