Dynamic Ranking and Translation Synchronization
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 translation synchronization problem, to the dynamic setting. In this setup, we are given a sequence of comparison graphs (G_t)_tβπ―, where π―β [0,1] is a grid representing the time domain, and for each item i and time tβπ― there is an associated unknown strength parameter z^*_t,iββ. We aim to recover, for tβπ―, the strength vector z^*_t=(z^*_t,1,β¦,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 β 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 β_2 estimation error under the assumption that G_t is connected for all tβπ―, thus proving the consistency of the proposed methods in terms of the grid size |π―|. We complement our theoretical findings with experiments on synthetic and real data.
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