  In many applications, such as sport tournaments or recommendation systems, we
have at our disposal data consisting of pairwise comparisons between a set of
$n$ items (or players). The objective is to use this data to infer the latent
strength of each item and/or their ranking. Existing results for this problem
predominantly focus on the setting consisting of a single comparison graph $G$.
However, there exist scenarios (e.g., sports tournaments) where the the
pairwise comparison data evolves with time. Theoretical results for this
dynamic setting are relatively limited and is the focus of this paper.
  We study an extension of the \emph{translation synchronization} problem, to
the dynamic setting. In this setup, we are given a sequence of comparison
graphs $(G_t)_{t\in \mathcal{T}}$, where $\mathcal{T} \subset [0,1]$ is a grid
representing the time domain, and for each item $i$ and time $t\in \mathcal{T}$
there is an associated unknown strength parameter $z^*_{t,i}\in \mathbb{R}$. We
aim to recover, for $t\in\mathcal{T}$, the strength vector
$z^*_t=(z^*_{t,1},\dots,z^*_{t,n})$ from noisy measurements of
$z^*_{t,i}-z^*_{t,j}$, where $\{i,j\}$ is an edge in $G_t$. Assuming that
$z^*_t$ evolves smoothly in $t$, we propose two estimators -- one based on a
smoothness-penalized least squares approach and the other based on projection
onto the low frequency eigenspace of a suitable smoothness operator. For both
estimators, we provide finite sample bounds for the $\ell_2$ estimation error
under the assumption that $G_t$ is connected for all $t\in \mathcal{T}$, thus
proving the consistency of the proposed methods in terms of the grid size
$|\mathcal{T}|$. We complement our theoretical findings with experiments on
synthetic and real data.
