# American Institute of Mathematical Sciences

August  2013, 7(3): 907-926. doi: 10.3934/ipi.2013.7.907

## Statistical ranking using the $l^{1}$-norm on graphs

 1 Department of Mathematics, University of California, Los Angeles 90095, United States, United States 2 Department of Mathematics, UCLA, Los Angeles, CA 90095-1555

Received  January 2012 Revised  January 2013 Published  September 2013

We consider the problem of establishing a statistical ranking for a set of alternatives from a dataset which consists of an (inconsistent and incomplete) set of quantitative pairwise comparisons of the alternatives. If we consider the directed graph where vertices represent the alternatives and the pairwise comparison data is a function on the arcs, then the statistical ranking problem is to find a potential function, defined on the vertices, such that the gradient of the potential optimally agrees with the pairwise comparisons. Potentials, optimal in the $l^{2}$-norm sense, can be found by solving a least-squares problem on the digraph and, recently, the residual has been interpreted using the Hodge decomposition (Jiang et. al., 2010). In this work, we consider an $l^{1}$-norm formulation of the statistical ranking problem. We describe a fast graph-cut approach for finding $\epsilon$-optimal solutions, which has been used successfully in image processing and computer vision problems. Applying this method to several datasets, we demonstrate its efficacy at finding solutions with sparse residual.
Citation: Braxton Osting, Jérôme Darbon, Stanley Osher. Statistical ranking using the $l^{1}$-norm on graphs. Inverse Problems and Imaging, 2013, 7 (3) : 907-926. doi: 10.3934/ipi.2013.7.907
##### References:
 [1] J.-P. Aubin and A. Cellina, "Differential Inclusions. Set-Valued Maps and Viability Theory," Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 264, Springer-Verlag, Berlin, 1984. doi: 10.1007/978-3-642-69512-4. [2] I. Ali, W. D. Cook and M. Kress, Ordinal ranking and intensity of preference: A linear programming approach, Management Science, 32 (1986), 1642-1647. doi: 10.1287/mnsc.32.12.1642. [3] R. Ahuja, D. Hochbaum and J. Orlin, Solving the convex cost integer dual network flow problem, Management Science, 49 (2003), 950-964. [4] R. K Ahuja, T. L. Magnanti and J. B. Orlin, "Network Flows: Theory, Algorithms and Applications," Prentice Hall, Inc., Englewood Cliffs, NJ, 1993. [5] , 2011 ATP world tour media guide, accessed 10/5/2011. Available from: http://www.atpworldtour.com/Press/Rankings-and-Stats.aspx. [6] J. M. Bioucas-Dias and G. Valadao, Phase unwrapping via graph cuts, IEEE Transactions on Image Processing, 16 (2007), 698-709. doi: 10.1109/TIP.2006.888351. [7] Y. Boykov and V. Kolmogorov, An experimental comparison of min-cut/max-flow algorithms for energy minimization in vision, IEEE Transactions on Pattern Analysis and Machine Intelligence, 26 (2004), 1124-1137. [8] Y. Boykov, O. Veksler and R. Zabih, Fast approximate energy minimization via graph cuts, IEEE Transactions on Pattern Analysis and Machine Intelligence, 23 (2001), 1222-1239. [9] I. Charon and O. Hudry, A survey on the linear ordering problem for weighted or unweighted tournaments, 4OR, 5 (2007), 5-60. doi: 10.1007/s10288-007-0036-6. [10] W. D. Cook and M. Kress, Ordinal ranking with intensity preference, Management Science, 31 (1985), 26-32. doi: 10.1287/mnsc.31.1.26. [11] E. Candes and T. Tao, Near optimal signal recovery from random projections: Univeral encoding strategies?, IEEE Transactions on Information Theory, 52 (2006), 5406-5425. doi: 10.1109/TIT.2006.885507. [12] J. Darbon, "Composants Logiciels et Algorithmes de Minimisation Exacte d'Énergies dÉdiés au Traitement des Images," Ph.D. thesis, Ecole Nationale Supérieure des Télćommunications, October, 2005. [13] H. A. David, "The Method of Paired Comparisons," Hafner Publishing Co., New York, 1963. [14] , C. Dwork, R. Kumar, M. Naor and D. Sivakumar, Rank aggregation revisited. [15] _______, Rank aggregation methods for the web, Proceedings International Conference World Wide Web (WWW'01), 10 (2001), 613-622. [16] L. R. Foulds, "Graph Theory Applications," Universitext, Springer-Verlag, New York, 1992. doi: 10.1007/978-1-4612-0933-1. [17] M. Grant and S. Boyd, Graph implementations for nonsmooth convex programs, in "Recent Advances in Learning and Control" (eds. V. Blondel, S. Boyd and H. Kimura), Lecture Notes in Control and Information Sciences, 371, Springer, London, (2008), 95-110. Available from: http://stanford.edu/ boyd/graph_dcp.html. doi: 10.1007/978-1-84800-155-8_7. [18] ______, CVX: Matlab software for disciplined convex programming, version 1.21, April, 2011. Available from: http://cvxr.com/cvx. [19] D. Goldfarb and G. Iyengar, Robust portfolio selection problems, Mathematics of Operations Research, 28 (2003), 1-38. doi: 10.1287/moor.28.1.1.14260. [20] D. F. Gleich and L.-H. Lim, Rank aggregation via nuclear norm minimization, in "Proceedings of the 17th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining," ACM, New York, (2011), 60-68. doi: 10.1145/2020408.2020425. [21] D. Harville, The use of linear-model methodology to rate high school or college football teams, Journal of the American Statistical Association, 72 (1977), 278-289. [22] A. N. Hirani, K. Kalyanaraman and S. Watts, Least squares ranking on graphs, arXiv:1011.1716v4, (2011). [23] D. S. Hochbaum and E. Moreno-Centeno, Country credit-risk rating aggregation via the separation-deviation model, Optimization Methods and Software, 23 (2008), 741-762. doi: 10.1080/10556780802402432. [24] D. S. Hochbaum, The separation and separation-deviation methodology for group decision making and aggregate ranking, TutORials in Operations Research, 7 (2010), 116-141. [25] , Jester: The online joke recommender, accessed 9/9/2011. Available from: http://eigentaste.berkeley.edu/dataset/jester_dataset_2.zip. [26] X. Jiang, L.-H. Lim, Y. Yao and Y. Ye, Statistical ranking and combinatorial Hodge theory, Math. Program. Ser. B, 127 (2011), 203-244. doi: 10.1007/s10107-010-0419-x. [27] J. G. Kemeny and J. L. Snell, "Mathematical Models in the Social Sciences," The MIT Press, 1962. [28] V. Kolmogorov and A. Shioura, New algorithms for convex cost tension problem with application to computer vision, Discrete Optimization, 6 (2009), 378-393. doi: 10.1016/j.disopt.2009.04.006. [29] V. Kolmogorov and R. Zabih, What energy functions can be minimized via graph cuts?, IEEE Transactions on Pattern Analysis and Machine Intelligence, 2 (2004), 147-159. [30] J. I. Marden, "Analyzing and Modeling Rank Data," Monographs on Statistics and Applied Probability, 64, Chapman & Hall, London, 1995. [31] W. Ma, J.-M. Morel, S. Osher and A. Chien, An l1-based model for retinex theory and its application to medical images, IEEE Conference on Computer Vision and Pattern Recognition, (2011), 153-160. [32] K. Murota, "Discrete Convex Optimization," SIAM Society for Industrial and Applied Mathematics, 2003. [33] R. T. Rockafellar, "Convex Analysis," Princeton Mathematical Series, No. 28, Princeton University Press, Princeton, NJ, 1970. [34] R. T. Rockafellar, "Network Flows and Monotropic Optimization," Pure and Applied Mathematics (New York), A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1984. [35] D. G. Saari, Mathematical structure of voting paradoxes. I. Pairwise votes, Economic Theory, 15 (2000), 1-53. doi: 10.1007/s001990050001. [36] , Scrapy web crawling framework, accessed 10/5/2011. Available from: http://scrapy.org/. [37] R. T. Stefani, Football and basketball predictions using least squares, IEEE Transactions on Systems, Man, and Cybernetics, 7 (1977), 117-121. [38] , Tennis datenbank website, accessed 10/5/2011. Available from: http://tennis.wettpoint.com/en/. [39] N. M. Tran, Pairwise ranking: Choice of method can produce arbitrarily different rank order, Linear Algebra Appl., 438 (2013), 1012-1024. doi: 10.1016/j.laa.2012.08.028. [40] _______, Hodgerank is the limit of Perron Rank, preprint, arXiv:1201.4632, (2012). [41] Q. Xu, Y. Yao, T. Jiang, Q. Huang, B. Yan and W. Lin, Random partial paired comparison for subjective video quality assessment via hodgerank, in "Proceedings of the 19th ACM International Conference on Multimedia," ACM, New York, (2011), 393-402. doi: 10.1145/2072298.2072350. [42] , Yahoo! Webscope dataset: ydata-ymovies-user-movie-ratings-content-v1_0, accessed 10/5/2011. Available from: http://webscope.sandbox.yahoo.com. [43] H. P. Young, Condorcet's theory of voting, The American Political Science Review, 82 (1988), 1231-1244.

show all references

##### References:
 [1] J.-P. Aubin and A. Cellina, "Differential Inclusions. Set-Valued Maps and Viability Theory," Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 264, Springer-Verlag, Berlin, 1984. doi: 10.1007/978-3-642-69512-4. [2] I. Ali, W. D. Cook and M. Kress, Ordinal ranking and intensity of preference: A linear programming approach, Management Science, 32 (1986), 1642-1647. doi: 10.1287/mnsc.32.12.1642. [3] R. Ahuja, D. Hochbaum and J. Orlin, Solving the convex cost integer dual network flow problem, Management Science, 49 (2003), 950-964. [4] R. K Ahuja, T. L. Magnanti and J. B. Orlin, "Network Flows: Theory, Algorithms and Applications," Prentice Hall, Inc., Englewood Cliffs, NJ, 1993. [5] , 2011 ATP world tour media guide, accessed 10/5/2011. Available from: http://www.atpworldtour.com/Press/Rankings-and-Stats.aspx. [6] J. M. Bioucas-Dias and G. Valadao, Phase unwrapping via graph cuts, IEEE Transactions on Image Processing, 16 (2007), 698-709. doi: 10.1109/TIP.2006.888351. [7] Y. Boykov and V. Kolmogorov, An experimental comparison of min-cut/max-flow algorithms for energy minimization in vision, IEEE Transactions on Pattern Analysis and Machine Intelligence, 26 (2004), 1124-1137. [8] Y. Boykov, O. Veksler and R. Zabih, Fast approximate energy minimization via graph cuts, IEEE Transactions on Pattern Analysis and Machine Intelligence, 23 (2001), 1222-1239. [9] I. Charon and O. Hudry, A survey on the linear ordering problem for weighted or unweighted tournaments, 4OR, 5 (2007), 5-60. doi: 10.1007/s10288-007-0036-6. [10] W. D. Cook and M. Kress, Ordinal ranking with intensity preference, Management Science, 31 (1985), 26-32. doi: 10.1287/mnsc.31.1.26. [11] E. Candes and T. Tao, Near optimal signal recovery from random projections: Univeral encoding strategies?, IEEE Transactions on Information Theory, 52 (2006), 5406-5425. doi: 10.1109/TIT.2006.885507. [12] J. Darbon, "Composants Logiciels et Algorithmes de Minimisation Exacte d'Énergies dÉdiés au Traitement des Images," Ph.D. thesis, Ecole Nationale Supérieure des Télćommunications, October, 2005. [13] H. A. David, "The Method of Paired Comparisons," Hafner Publishing Co., New York, 1963. [14] , C. Dwork, R. Kumar, M. Naor and D. Sivakumar, Rank aggregation revisited. [15] _______, Rank aggregation methods for the web, Proceedings International Conference World Wide Web (WWW'01), 10 (2001), 613-622. [16] L. R. Foulds, "Graph Theory Applications," Universitext, Springer-Verlag, New York, 1992. doi: 10.1007/978-1-4612-0933-1. [17] M. Grant and S. Boyd, Graph implementations for nonsmooth convex programs, in "Recent Advances in Learning and Control" (eds. V. Blondel, S. Boyd and H. Kimura), Lecture Notes in Control and Information Sciences, 371, Springer, London, (2008), 95-110. Available from: http://stanford.edu/ boyd/graph_dcp.html. doi: 10.1007/978-1-84800-155-8_7. [18] ______, CVX: Matlab software for disciplined convex programming, version 1.21, April, 2011. Available from: http://cvxr.com/cvx. [19] D. Goldfarb and G. Iyengar, Robust portfolio selection problems, Mathematics of Operations Research, 28 (2003), 1-38. doi: 10.1287/moor.28.1.1.14260. [20] D. F. Gleich and L.-H. Lim, Rank aggregation via nuclear norm minimization, in "Proceedings of the 17th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining," ACM, New York, (2011), 60-68. doi: 10.1145/2020408.2020425. [21] D. Harville, The use of linear-model methodology to rate high school or college football teams, Journal of the American Statistical Association, 72 (1977), 278-289. [22] A. N. Hirani, K. Kalyanaraman and S. Watts, Least squares ranking on graphs, arXiv:1011.1716v4, (2011). [23] D. S. Hochbaum and E. Moreno-Centeno, Country credit-risk rating aggregation via the separation-deviation model, Optimization Methods and Software, 23 (2008), 741-762. doi: 10.1080/10556780802402432. [24] D. S. Hochbaum, The separation and separation-deviation methodology for group decision making and aggregate ranking, TutORials in Operations Research, 7 (2010), 116-141. [25] , Jester: The online joke recommender, accessed 9/9/2011. Available from: http://eigentaste.berkeley.edu/dataset/jester_dataset_2.zip. [26] X. Jiang, L.-H. Lim, Y. Yao and Y. Ye, Statistical ranking and combinatorial Hodge theory, Math. Program. Ser. B, 127 (2011), 203-244. doi: 10.1007/s10107-010-0419-x. [27] J. G. Kemeny and J. L. Snell, "Mathematical Models in the Social Sciences," The MIT Press, 1962. [28] V. Kolmogorov and A. Shioura, New algorithms for convex cost tension problem with application to computer vision, Discrete Optimization, 6 (2009), 378-393. doi: 10.1016/j.disopt.2009.04.006. [29] V. Kolmogorov and R. Zabih, What energy functions can be minimized via graph cuts?, IEEE Transactions on Pattern Analysis and Machine Intelligence, 2 (2004), 147-159. [30] J. I. Marden, "Analyzing and Modeling Rank Data," Monographs on Statistics and Applied Probability, 64, Chapman & Hall, London, 1995. [31] W. Ma, J.-M. Morel, S. Osher and A. Chien, An l1-based model for retinex theory and its application to medical images, IEEE Conference on Computer Vision and Pattern Recognition, (2011), 153-160. [32] K. Murota, "Discrete Convex Optimization," SIAM Society for Industrial and Applied Mathematics, 2003. [33] R. T. Rockafellar, "Convex Analysis," Princeton Mathematical Series, No. 28, Princeton University Press, Princeton, NJ, 1970. [34] R. T. Rockafellar, "Network Flows and Monotropic Optimization," Pure and Applied Mathematics (New York), A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1984. [35] D. G. Saari, Mathematical structure of voting paradoxes. I. Pairwise votes, Economic Theory, 15 (2000), 1-53. doi: 10.1007/s001990050001. [36] , Scrapy web crawling framework, accessed 10/5/2011. Available from: http://scrapy.org/. [37] R. T. Stefani, Football and basketball predictions using least squares, IEEE Transactions on Systems, Man, and Cybernetics, 7 (1977), 117-121. [38] , Tennis datenbank website, accessed 10/5/2011. Available from: http://tennis.wettpoint.com/en/. [39] N. M. Tran, Pairwise ranking: Choice of method can produce arbitrarily different rank order, Linear Algebra Appl., 438 (2013), 1012-1024. doi: 10.1016/j.laa.2012.08.028. [40] _______, Hodgerank is the limit of Perron Rank, preprint, arXiv:1201.4632, (2012). [41] Q. Xu, Y. Yao, T. Jiang, Q. Huang, B. Yan and W. Lin, Random partial paired comparison for subjective video quality assessment via hodgerank, in "Proceedings of the 19th ACM International Conference on Multimedia," ACM, New York, (2011), 393-402. doi: 10.1145/2072298.2072350. [42] , Yahoo! Webscope dataset: ydata-ymovies-user-movie-ratings-content-v1_0, accessed 10/5/2011. Available from: http://webscope.sandbox.yahoo.com. [43] H. P. Young, Condorcet's theory of voting, The American Political Science Review, 82 (1988), 1231-1244.
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