Dynamic ranking and translation synchronization

Abstract

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 these 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 pairwise comparison data evolve with time. Theoretical results for this dynamic setting are relatively limited, and are the focus of this paper. We study an extension of the translation synchronization problem, to the dynamic setting. In this set-up, we are given a sequence of comparison graphs $(G_t)_{t\in{{\mathscr{T}}}}$, where $ {{\mathscr{T}}} \subset [0,1]$ is a grid representing the time domain, and for each item $i$ and time $t\in{{\mathscr{T}}}$ there is an associated unknown strength parameter $z^*_{t,i}\in{{\mathbb{R}}}$. We aim to recover, for $t\in{{\mathscr{T}}}$, the strength vector $z^*_t=(z^*_{t,1},\dots ,z^*_{t,n})$ from noisy measurements of $z^*_{t,i}-z^*_{t,j}$, where $\left \{{i,j}\right \}$ 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 $t\in{{\mathscr{T}}}$, thus proving the consistency of the proposed methods in terms of the grid size $|\mathscr{T}|$. We complement our theoretical findings with experiments on synthetic and real data.