[Paper Review] Ranking and synchronization from pairwise measurements via SVD

Alexandre d’Aspremont, Mihai Cucuringu+1 more|arXiv (Cornell University)|Jun 6, 2019

Sparse and Compressive Sensing TechniquesEngineering53 references17 citations

TL;DR

This paper proposes SVD-RS and SVD-NRS, spectral algorithms based on singular value decomposition (SVD) for robustly recovering rankings and synchronizing signals from noisy pairwise measurements. The methods achieve strong theoretical guarantees in $ε$-recovery and rank consistency under sparse and noisy sampling, with empirical performance competitive against state-of-the-art methods on real-world data sets including sports and synchronization problems.

ABSTRACT

Given a measurement graph $G= (V,E)$ and an unknown signal $r \in \mathbb{R}^n$, we investigate algorithms for recovering $r$ from pairwise measurements of the form $r_i - r_j$; $\{i,j\} \in E$. This problem arises in a variety of applications, such as ranking teams in sports data and time synchronization of distributed networks. Framed in the context of ranking, the task is to recover the ranking of $n$ teams (induced by $r$) given a small subset of noisy pairwise rank offsets. We propose a simple SVD-based algorithmic pipeline for both the problem of time synchronization and ranking. We provide a detailed theoretical analysis in terms of robustness against both sampling sparsity and noise perturbations with outliers, using results from matrix perturbation and random matrix theory. Our theoretical findings are complemented by a detailed set of numerical experiments on both synthetic and real data, showcasing the competitiveness of our proposed algorithms with other state-of-the-art methods.

Motivation & Objective

• Address the challenge of recovering latent rankings or synchronization signals from sparse, noisy pairwise comparisons.
• Develop a unified spectral framework applicable to both ranking and time synchronization problems.
• Provide theoretical robustness guarantees against sampling sparsity and noise, particularly in the presence of outliers.
• Demonstrate strong empirical performance across diverse real-world data sets, including sports and wireless network synchronization.
• Extend the framework with matrix completion as a preprocessing step to improve recovery in low-sampling regimes.

Proposed method

• Formulate the problem as recovering a latent signal $ r \in \mathbb{R}^n $ from noisy pairwise differences $ r_i - r_j $ on a graph $ G = (V, E) $.
• Construct a skew-symmetric matrix $ H $ encoding the pairwise comparisons $ r_i - r_j $, then apply SVD to the matrix $ H $ to extract the dominant eigenvector.
• Propose SVD-RS (SVD-based Ranking and Synchronization) and its normalized variant SVD-NRS, where normalization accounts for graph degree distribution.
• Use matrix completion as a preprocessing step to impute missing pairwise comparisons, enhancing performance on sparse graphs.
• Apply data-driven procedures to estimate the global scale (e.g., fixing the root node) after SVD-based recovery.
• Leverage tools from matrix perturbation theory and random matrix theory to derive theoretical bounds on recovery error.

Experimental results

Research questions

• RQ1 Can SVD-based methods achieve consistent recovery of the latent signal $ r $ under sparse and noisy pairwise measurements?
• RQ2 How does the performance of SVD-RS and SVD-NRS compare to state-of-the-art methods in terms of $ \ell_2 $ and $ \ell_\infty $ error bounds?
• RQ3 What is the theoretical sample complexity required for $ \ell_2 $ and $ \ell_\infty $ recovery in the presence of noise and sparsity?
• RQ4 Does incorporating low-rank matrix completion as a preprocessing step improve recovery accuracy, especially in sparse regimes?
• RQ5 Can the $ \ell_\infty $ error of SVD-RS be used to derive guarantees on rank recovery, such as maximum displacement error?

Key findings

• For SVD-RS, $ \Omega(n \log n) $ measurements suffice for $ \ell_2 $ recovery of the score vector $ r $, and $ \Omega(n^{4/3} (\log n)^{2/3}) $ for $ \ell_\infty $ recovery.
• The $ \ell_\infty $ analysis of SVD-RS leads to a bound on the maximum displacement error in rank recovery, ensuring consistent ranking under noise.
• SVD-NRS achieves $ \ell_2 $ recovery guarantees under the same theoretical framework, with improved robustness due to normalization.
• On the Microsoft Halo 2 Beta data set, SVD-N outperformed SVD in three out of four performance metrics and ranked in the top 3 among nine algorithms.
• In the Premier League data set (2009–2013), SVD-N outperformed SVD in 12 out of 16 performance evaluations and ranked first in three instances.
• The matrix completion preprocessing step significantly improved recovery performance on sparse graphs, particularly in reducing upsets and weighted upsets.