Foreword to the Special Focus on Mathematics, Data and Knowledge


There is a growing interest in applying mathematical theories and methods from topology, computational geometry, differential equations, fluid dynamics, quantum statistics, etc. to describe and to analyze scientific regularities of diverse, massive, complex, nonlinear, and fast changing data accumulated continuously around the world and in discovering and revealing valid, insightful, and valuable knowledge that data imply. With increasingly solid mathematical foundations, various methods and techniques have been studied and developed for data mining, modeling, and processing, and knowledge representation, organization, and verification; different systems and mechanisms have been designed to perform data-intensive tasks in many application fields for classification, predication, recommendation, ranking, filtering, etc. This special focus of Mathematics in Computer Science is organized to stimulate original research on the interaction of mathematics with data and knowledge, in particular the exploration of new mathematical theories and methodologies for data modeling and analysis and knowledge discovery and management, the study of mathematical models of big data and complex knowledge, and the development of novel solutions and strategies to enhance the performance of existing systems and mechanisms for data and knowledge processing. The present foreword provides a short review of some key ideas and techniques on how mathematics interacts with data and knowledge, together with a few selected research directions and problems and a brief introduction to the four papers published in the focus.

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


  1. 1

    Amari S.-I.: Information geometry on hierarchy of probability distributions. IEEE Trans. Inf. Theory 47(5), 1701–1711 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  2. 2

    Amari S.-I., Nagaoka H.: Methods of Information Geometry. American Mathematical Society, Providence, RI (2000)

    MATH  Google Scholar 

  3. 3

    Arotaritei D., Mitra S.: Web mining: a survey in the fuzzy framework. Fuzzy Sets Syst. 148(1), 5–19 (2004)

    Article  MathSciNet  Google Scholar 

  4. 4

    Bennett K.P., Parrado-Hernandez E.: The interplay of optimization and machine learning research. J. Mach. Learn. Res. 7, 1265–1281 (2006)

    MATH  MathSciNet  Google Scholar 

  5. 5

    Bezdek J.C.: Pattern Recognition with Fuzzy Objective Function Algorithms. Plenum Press, New York (1981)

    Book  MATH  Google Scholar 

  6. 6

    Bishop C.M.: Pattern Recognition and Machine Learning. Springer, New York (2006)

    MATH  Google Scholar 

  7. 7

    Bradley P.S., Fayyad U.M., Mangasarian O.L.: Mathematical programming for data mining: formulations and challenges. INFORMS J. Comput. 11(3), 217–238 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  8. 8

    Burges C.J.C.: A tutorial on support vector machines for pattern recognition. Data Min. Knowl. Discov. 2(2), 121–167 (1998)

    Article  Google Scholar 

  9. 9

    Carette, J., Farmer, W.M.: A review of mathematical knowledge management. In: Carette, J., Dixon, L., Coen, C.S., Watt, S.M. (eds.) Intelligent Computer Mathematics, volume 5625 of Lecture Notes in Artificial Intelligence, pp. 233–246. Springer, Berlin (2009)

  10. 10

    Carlsson, G.: Topology and data. Bull. Am. Math. Soc. 46(2), 255–308 (2009)

    Google Scholar 

  11. 11

    Chazal F., Cohen-Steiner D., Mrigot Q.: Geometric inference for probability measures. Found. Comput. Math. 11(6), 733–751 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  12. 12

    Chen X., Wang D.: Management of geometric knowledge in textbooks. Data Knowl. Eng. 73, 43–57 (2012)

    Article  Google Scholar 

  13. 13

    Chin, F.Y., Leung, H.C.: Voting algorithms for discovering long motifs. In: Chen, Y.-P.P., Wong, L. (eds.) Proceedings of the 3rd Asia-Pacific Bioinformatics Conference, volume 1 of Series on Advances in Bioinformatics and Computational Biology, pp. 261–271. World Scientific Publishing Company, Singapore (2005)

  14. 14

    d’Aspremont A., Ghaoui L.E., Jordan M.I., Lanckriet G.R.G.: A direct formulation for sparse PCA using semidefinite programming. SIAM Rev. 49(3), 434–448 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  15. 15

    Devore J.L., Berk K.N.: Modern Mathematical Statistics with Applications. Springer, New York (2007)

    MATH  Google Scholar 

  16. 16

    Diao, Y., Li, B., Liu, A., Peng, L., Sutton, C.A., Tran, T.T.L., Zink, M.: Capturing data uncertainty in high-volume stream processing. In: Proceedings of the 4th Biennial Conference on Innovative Data Systems Research, Asilomar, California, USA, January 4–7, 2009. (Online Proceedings)

  17. 17

    Dunn J.C.: A fuzzy relative of the ISODATA process and its use in detecting compact well-separated clusters. J. Cybern. 3, 32–57 (1973)

    MATH  MathSciNet  Google Scholar 

  18. 18

    Friedman, M., Last, M., Zaafrany, O., Schneider, M., Kandel, A.: A new approach for fuzzy clustering of web documents. In: Proceedings of IEEE International Conference on Fuzzy Systems, volume 1, pp. 377–381. IEEE Computational Intelligence Society (2004)

  19. 19

    Ghrist R.: Barcodes: the persistent topology of data. Bull. Am. Meteorol. Soc. 45, 61–75 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  20. 20

    Kerber, M. (ed.): Management of Mathematical Knowledge—Special issue of Mathematics in Computer Science, vol 2. Birkhäuser, Basel (2008)

  21. 21

    Lian, X., Chen, L.: A generic framework for handling uncertain data with local correlations. Proc. VLDB Endow. 4(1), 12–21 (2010)

    Google Scholar 

  22. 22

    Ropero J., Leun C., Carrasco A., Gumez A., Rivera O.: Fuzzy logic applications for knowledge discovery: a survey. Int. J. Adv. Comput. Technol. 3(6), 187–198 (2011)

    Google Scholar 

  23. 23

    Roweis S.T., Saul L.K.: Nonlinear dimensionality reduction by locally linear embedding. Science 290(5500), 2323–2326 (2000)

    Article  Google Scholar 

  24. 24

    Scheinberg, K., Peng, J., Terlaky, T., Shuurmans, D., Jordan, M., Poggio, T.: Mathematical programming in machine learning and data mining. In: 5-day Worksop Held by Banff International Research Station for Mathematical Innovation and Discovery, January 14–19 (2007)

  25. 25

    Silva, V.D., Carlsson, G.: Topological estimation using witness complexes. In: Alexa, M., Gross, M., Pfister, H., Rusinkiewicz, S. (eds.) Proceedings of the First Eurographics Conference on Point-Based Graphics, pp. 157–166. Eurographics Association Aire-la-Ville (2004)

  26. 26

    Simovici, D.A., Djeraba, C.: Mathematical Tools for Data Mining: Set Theory, Partial Orders, Combinatorics. Advanced Information and Knowledge Processing. Springer, London (2008)

  27. 27

    Tenenbaum J.B., De Silva V., Langford J.C.: A global geometric framework for nonlinear dimensionality reduction. Science 290(5500), 2319–2323 (2000)

    Article  Google Scholar 

  28. 28

    Tran T.T., Peng L., Diao Y., McGregor A., Liu A.: CLARO: modeling and processing uncertain data streams. VLDB J. 21(5), 651–676 (2012)

    Article  Google Scholar 

  29. 29

    Trendafilov N., Jolliffe I.T., Uddin M.: A modified principal component technique based on the LASSO. J. Comput. Graph. Stat. 12, 531–547 (2003)

    Article  MathSciNet  Google Scholar 

  30. 30

    Zou H., Hastie T., Tibshirani R.: Sparse principal component analysis. J. Comput. Graph. Stat. 15(2), 265–286 (2006)

    Article  MathSciNet  Google Scholar 

Download references

Author information



Corresponding author

Correspondence to Xiaoyu Chen.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Chen, X., Wang, D. & Zhang, X. Foreword to the Special Focus on Mathematics, Data and Knowledge. Math.Comput.Sci. 7, 379–386 (2013).

Download citation