Skip to main content

A Ranking Model Motivated by Nonnegative Matrix Factorization with Applications to Tennis Tournaments

Part of the Lecture Notes in Computer Science book series (LNAI,volume 11908)


We propose a novel ranking model that combines the Bradley-Terry-Luce probability model with a nonnegative matrix factorization framework to model and uncover the presence of latent variables that influence the performance of top tennis players. We derive an efficient, provably convergent, and numerically stable majorization-minimization-based algorithm to maximize the likelihood of datasets under the proposed statistical model. The model is tested on datasets involving the outcomes of matches between 20 top male and female tennis players over 14 major tournaments for men (including the Grand Slams and the ATP Masters 1000) and 16 major tournaments for women over the past 10 years. Our model automatically infers that the surface of the court (e.g., clay or hard court) is a key determinant of the performances of male players, but less so for females. Top players on various surfaces over this longitudinal period are also identified in an objective manner.


  • BTL ranking model
  • Nonnegative matrix factorization
  • Low-rank approximation
  • Majorization-minimization
  • Sports analytics

This is a preview of subscription content, access via your institution.

Buying options

USD   29.95
Price excludes VAT (USA)
  • DOI: 10.1007/978-3-030-46133-1_12
  • Chapter length: 17 pages
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
USD   89.00
Price excludes VAT (USA)
  • ISBN: 978-3-030-46133-1
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book
USD   119.99
Price excludes VAT (USA)
Fig. 1.


  1. 1.

    The updates for \(\mathbf {W}\) and \(\mathbf {H}\) are not symmetric because the data is in the form of a 3-way tensor \(\{b_{ij}^{(m)}\}\); this is also apparent in (1) and the updates in ().

  2. 2.

    One might be tempted to normalize . This, however, does not resolve numerical issues as some entries of may be zero.

  3. 3.

    This may be attributed to its position in the seasonal calendar. The Paris Masters is the last tournament before ATP World Tour Finals. Top players often choose to skip this tournament to prepare for the more prestigious ATP World Tour Finals. This has led to some surprising results, e.g., Ferrer, a strong clay player, won the Paris Masters in 2012 (even though the Paris Masters is a hard court tournament).

  4. 4.

    The solution with the highest likelihood is shown in Trial 2 of Table S-13 but it appears that the solution there is degenerate.

  5. 5.

    Stationary points are not necessarily equivalent up to permutation or rescaling.


  1. Elo, A.E.: The Rating of Chess Players, Past and Present. Ishi Press International, Bronx (2008)

    Google Scholar 

  2. Bradley, R., Terry, M.: Rank analysis of incomplete block designs I: the method of paired comparisons. Biometrika 35, 324–345 (1952)

    MathSciNet  MATH  Google Scholar 

  3. Luce, R.: Individual Choice Behavior: A Theoretical Analysis. Wiley, New York (1959)

    MATH  Google Scholar 

  4. Lee, D.D., Seung, H.S.: Learning the parts of objects with nonnegative matrix factorization. Nature 401, 788–791 (1999)

    CrossRef  Google Scholar 

  5. Cichocki, A., Zdunek, R., Phan, A.H., Amari, S.-I.: Nonnegative Matrix and Tensor Factorizations: Applications to Exploratory Multi-Way Data Analysis and Blind Source Separation. John Wiley & Sons Ltd., Chichester (2009)

    CrossRef  Google Scholar 

  6. Marden, J.I.: Analyzing and Modeling Rank Data. CRC Press, London (1996)

    MATH  Google Scholar 

  7. Lee, D.D., Seung, H.S.: Algorithms for nonnegative matrix factorization. In: Neural Information Processing Systems, pp. 535–541 (2000)

    Google Scholar 

  8. Févotte, C., Bertin, N., Durrieu, J.L.: Nonnegative matrix factorization with the Itakura-Saito divergence with application to music analysis. Neural Comput. 21(3), 793–830 (2009)

    CrossRef  Google Scholar 

  9. Berry, M.W., Browne, M.: Email surveillance using non-negative matrix factorization. Comput. Math. Organ. Theory 11(3), 249–264 (2005)

    CrossRef  Google Scholar 

  10. Geerts, A., Decroos, T., Davis, J.: Characterizing soccer players’ playing style from match event streams. In: Machine Learning and Data Mining for Sports Analytics ECML/PKDD 2018 Workshop, pp. 115–126 (2018)

    Google Scholar 

  11. Hunter, D.-R., Lange, K.: A tutorial on MM algorithms. Am. Stat. 58, 30–37 (2004)

    CrossRef  MathSciNet  Google Scholar 

  12. Zhao, R., Tan, V.Y.F.: A unified convergence analysis of the multiplicative update algorithm for regularized nonnegative matrix factorization. IEEE Trans. Signal Process. 66(1), 129–138 (2018)

    CrossRef  MathSciNet  Google Scholar 

  13. Razaviyayn, M., Hong, M., Luo, Z.Q.: A unified convergence analysis of block successive minimization methods for nonsmooth optimization. SIAM J. Optim. 23(2), 1126–1153 (2013)

    CrossRef  MathSciNet  Google Scholar 

  14. Oh, S., Shah, D.: Learning mixed multinomial logit model from ordinal data. In: Neural Information Processing Systems, pp. 595–603 (2014)

    Google Scholar 

  15. Shah, N.-B., Wainwright, M.-J.: Simple, robust and optimal ranking from pairwise comparisons. J. Mach. Learn. Res. 18(199), 1–38 (2018)

    MathSciNet  MATH  Google Scholar 

  16. Suh, C., Tan, V.Y.F., Zhao, R.: Adversarial top-\(K\) ranking. IEEE Trans. Inf. Theory 63(4), 2201–2225 (2017)

    CrossRef  MathSciNet  Google Scholar 

  17. Ding, W., Ishwar, P., Saligrama, V.: A topic modeling approach to ranking. In: Proceedings of the 18th International Conference on Artificial Intelligence and Statistics (AISTATS), pp. 214–222 (2015)

    Google Scholar 

  18. Févotte, C., Idier, J.: Algorithms for nonnegative matrix factorization with the \(\beta \)-divergence. Neural Comput. 23(9), 2421–2456 (2011)

    CrossRef  MathSciNet  Google Scholar 

  19. Tan, V.Y.F., Févotte, C.: Automatic relevance determination in nonnegative matrix factorization with the \(\beta \)-divergence. IEEE Trans. Pattern Anal. Mach. Intell. 35(7), 1592–1605 (2013)

    CrossRef  Google Scholar 

  20. Hunter, D.R.: MM algorithms for generalized Bradley-Terry models. Ann. Stat. 32(1), 384–406 (2004)

    CrossRef  MathSciNet  Google Scholar 

  21. Xia, R., Tan, V.Y.F., Filstroff, L., Févotte, C.: Supplementary material for “A ranking model motivated by nonnegative matrix factorization with applications to tennis tournaments” (2019).

  22. Dempster, A.P., Laird, N.M., Rubin, D.B.: Maximum likelihood from incomplete data via the EM algorithm. J. R. Stat. Soc. B 39, 1–38 (1977)

    MathSciNet  MATH  Google Scholar 

  23. Caron, F., Doucet, A.: Efficient Bayesian inference for generalized Bradley-Terry models. J. Comput. Graph. Stat. 21(1), 174–196 (2012)

    CrossRef  MathSciNet  Google Scholar 

Download references


This work was supported by a Ministry of Education Tier 2 grant (R-263-000-C83-112), an NRF Fellowship (R-263-000-D02-281), and by the European Research Council (ERC FACTORY-CoG-6681839).

Author information

Authors and Affiliations


Corresponding author

Correspondence to Rui Xia .

Editor information

Editors and Affiliations

1 Electronic supplementary material

Below is the link to the electronic supplementary material.

Supplementary material 1 (pdf 448 KB)

Rights and permissions

Reprints and Permissions

Copyright information

© 2020 Springer Nature Switzerland AG

About this paper

Verify currency and authenticity via CrossMark

Cite this paper

Xia, R., Tan, V.Y.F., Filstroff, L., Févotte, C. (2020). A Ranking Model Motivated by Nonnegative Matrix Factorization with Applications to Tennis Tournaments. In: Brefeld, U., Fromont, E., Hotho, A., Knobbe, A., Maathuis, M., Robardet, C. (eds) Machine Learning and Knowledge Discovery in Databases. ECML PKDD 2019. Lecture Notes in Computer Science(), vol 11908. Springer, Cham.

Download citation

  • DOI:

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-46132-4

  • Online ISBN: 978-3-030-46133-1

  • eBook Packages: Computer ScienceComputer Science (R0)