Improved theoretical guarantee for rank aggregation via spectral method

TL;DR

This paper improves the theoretical understanding of spectral methods for rank aggregation under the Erdös-Rényi outliers model, providing sharper error bounds and reduced sample complexity requirements.

Contribution

It offers a novel, sharper $ ext{l}_{ ext{infinity}}$-norm eigenvector perturbation bound for spectral ranking algorithms, enhancing theoretical guarantees for rank recovery.

Findings

Sharper eigenvector perturbation bounds established

Reduced sample complexity to $ ext{Omega}(n ext{log} n)$

Numerical experiments confirm theoretical improvements

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. As it is generally NP-hard to find a global ranking that minimizes the mismatch (known as the Kemeny optimization), we focus on the Erd\"os-R\'enyi outliers (ERO) model for this ranking problem. Here, each pairwise comparison is a corrupted copy of the true score difference. We investigate 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 also derive an error bound on the maximum displacement for each item, with only $\Omega(n\log n)$ samples. Our theoretical analysis improves upon the state-of-the-art results in terms of sample complexity, and our numerical experiments confirm these theoretical findings.