Improved theoretical guarantee for rank aggregation via spectral method

Abstract

Abstract Given pairwise comparisons between multiple items, how to rank them so that the ranking matches the observations? This problem, known as rank aggregation, has found many applications in sports, recommendation systems and other web applications. We focus on the ranking problem under the Erdös–Rényi outliers model: only a subset of pairwise comparisons is observed, being either clean or corrupted copies of the true score differences. We investigate the spectral ranking algorithms that are based on unnormalized and normalized data matrices. The key is to understand their performance in recovering the underlying scores of each item from the observed data. This reduces to deriving an entry-wise perturbation error bound between the top eigenvectors of the unnormalized/normalized data matrix and its population counterpart. By using the leave-one-out technique, we provide a sharper $\ell _{\infty }$-norm perturbation bound of the eigenvectors and derive an error bound on the maximum displacement for each item, with only $O(n\log n)$ samples. In addition, we also derive the sample complexity to perform top-$K$ ranking under mild assumptions. Our theoretical analysis improves upon the state-of-the-art results in terms of sample complexity, and our numerical experiments confirm these theoretical findings.