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.

翻译：在许多应用场景中，例如体育锦标赛或推荐系统，我们可获得由$n$个条目（或选手）之间两两比较所构成的数据。目标在于利用这些数据推断每个条目的潜在强度及其排名。现有针对该问题的研究主要集中在单一比较图$G$的设置上。然而，存在（例如体育锦标赛中）成对比较数据随时间演化的场景。针对动态设置的理论结果相对有限，而这正是本文的研究重点。我们将\emph{翻译同步}问题拓展至动态场景。在该设置下，给定一系列比较图$(G_t)_{t\in \mathcal{T}}$，其中$\mathcal{T} \subset [0,1]$为表示时间域的网格，且对每个条目$i$及时间$t\in \mathcal{T}$，存在关联的未知强度参数$z^*_{t,i}\in \mathbb{R}$。我们的目标是通过对$z^*_{t,i}-z^*_{t,j}$（其中$\{i,j\}$为$G_t$中的边）的含噪测量值，恢复每个$t\in\mathcal{T}$下的强度向量$z^*_t=(z^*_{t,1},\dots,z^*_{t,n})$。假设$z^*_t$在$t$上平滑演化，我们提出两种估计器——一种基于平滑惩罚最小二乘方法，另一种基于投影至合适平滑算子的低频特征空间。对于两种估计器，我们在假设$G_t$对所有$t\in \mathcal{T}$均连通的前提下，给出了$\ell_2$估计误差的有限样本界，从而证明了所提方法在网格规模$|\mathcal{T}|$意义上的一致性。我们通过在合成数据与真实数据上的实验补充了理论发现。