Deeply rooted in classical social choice and voting theory, statistical ranking with paired comparison data experienced its renaissance with the wide spread of crowdsourcing technique. As the data quality might be significantly damaged in an uncontrolled crowdsourcing environment, outlier detection and robust ranking have become a hot topic in such data analysis. In this paper, we propose a robust ranking framework based on the principle of Huber's robust statistics, which formulates outlier detection as a LASSO problem to find sparse approximations of the cyclic ranking projection in Hodge decomposition. Moreover, simple yet scalable algorithms are developed based on Linearized Bregman Iteration to achieve an even less biased estimator than LASSO. Statistical consistency of outlier detection is established in both cases which states that when the outliers are strong enough and in Erdos-Renyi random graph sampling settings, outliers can be faithfully detected. Our studies are supported by experiments with both simulated examples and real-world data. The proposed framework provides us a promising tool for robust ranking with large scale crowdsourcing data arising from computer vision, multimedia, machine learning, sociology, etc.
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