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 $z^*_t=(z^*_{t,1},\cdots,z^*_{t,n})$ 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：在许多应用中,比如体育锦标赛或建议系统,我们拥有由一组美元项目(或玩家)之间的配对比较构成的处置数据。我们的目标是使用这些数据来推断每个项目和(或)其排名的潜在强度。这个问题的现有结果主要侧重于由单一比较图形($G$)组成的设置。然而,存在一些情景(例如体育锦标赛),配对比较数据随着时间而演变。这个动态设置的理论结果相对有限,并且是本文的重点。我们研究的是 美元(或)调调调调调调时问题的问题,到动态设置。在这个设置中,我们得到了一个比较图表(g_t) 美元(G_t) 的序列。 美元(g) 美元(t) 美元(t) 美元(x) 美元(x) 平流化数据(t) 美元(t) 基调价(t) 美元(t) 美元(t) 基调时, 基调(x) 美元(t) 美元(t) 美元(t) 美元(t) 美元(t) 美元(t) 美元(t) ======(t) mabl====) 底。