Effective signal reconstruction from multiple ranked lists via convex optimization

Abstract

The ranking of objects is widely used to rate their relative quality or relevance across multiple assessments. Beyond classical rank aggregation, it is of interest to estimate the usually unobservable latent signals that inform a consensus ranking. Under the only assumption of independent assessments, which can be incomplete, we introduce indirect inference via convex optimization in combination with computationally efficient Poisson Bootstrap. Two different objective functions are suggested, one linear and the other quadratic. The mathematical formulation of the signal estimation problem is based on pairwise comparisons of all objects with respect to their rank positions. Sets of constraints represent the order relations. The transitivity property of rank scales allows us to reduce substantially the number of constraints associated with the full set of object comparisons. The key idea is to globally reduce the errors induced by the rankers until optimal latent signals can be obtained. Its main advantage is low computational costs, even when handling n<<p\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n < < p$$\\end{document} data problems. Exploratory tools can be developed based on the bootstrap signal estimates and standard errors. Simulation evidence, a comparison with the state-of-the-art rank centrality method, and two applications, one in higher education evaluation and the other in molecular cancer research, are presented.