Data Mining and Knowledge Discovery

, Volume 26, Issue 3, pp 512–532 | Cite as

Dependence maps, a dimensionality reduction with dependence distance for high-dimensional data

  • Kichun Lee
  • Alexander Gray
  • Heeyoung Kim


We introduce the dependence distance, a new notion of the intrinsic distance between points, derived as a pointwise extension of statistical dependence measures between variables. We then introduce a dimension reduction procedure for preserving this distance, which we call the dependence map. We explore its theoretical justification, connection to other methods, and empirical behavior on real data sets.


Dependence maps Dimensionality reduction Dependence Markov chain 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Belkin M, Niyogi P (2003) Laplacian eigenmaps for dimensionality reduction and data representation. Neural Comput 15(6): 1373–1396zbMATHCrossRefGoogle Scholar
  2. Bottou L, Cortes C, Denker J, Drucker H, Guyon I, Jackel L, LeCun Y, Muller U, Sackinger E, Simard P, et al (1994) Comparison of classifier methods: a case study in handwritten digitrecognition. In: Proceedings of the 12th IAPR international. conference on pattern recognition, vol 2-conference B: computer vision & image processingGoogle Scholar
  3. Donoho DL, Grimes C (2003) Hessian eigenmaps: locally linear embedding techniques for high-dimensional data. Proc Natl Acad Sci USA 100(10): 5591–5596MathSciNetzbMATHCrossRefGoogle Scholar
  4. Haykin S (2008) Neural networks: a comprehensive foundation. Prentice Hall, Upper Saddle RiverGoogle Scholar
  5. Lafon S, Keller Y, Coifman RR (2006) Data fusion and multicue data matching by diffusion maps. IEEE Trans Pattern Anal Mach Intell 28(11):1784–1797. Google Scholar
  6. Lee K, Abouelnasr M, Bayer C, Gabram S, Mizaikoff B, Rogatko A, Vidakovic B (2009) Mining exhaled volatile organic compounds for breast cancer detection. Adv Appl Stat Sci 1: 327–342MathSciNetGoogle Scholar
  7. Mahadevan S, Maggioni M (2006) Value function approximation with diffusion wavelets and Laplacian eigenfunctions. Adv Neural Inf Process Syst 18: 843Google Scholar
  8. Mari D, Kotz S (2001) Correlation and dependence. Imperial College Press, LondonzbMATHCrossRefGoogle Scholar
  9. Nelsen R (2006) An introduction to copulas. Springer, New YorkzbMATHGoogle Scholar
  10. Roweis ST, Saul LK (2000) Nonlinear dimensionality reduction by locally linear embedding. Science 290(5500): 2323–2326. doi: 10.1126/science.290.5500.2323 CrossRefGoogle Scholar
  11. Scholkopf B, Smola A, Muller K (1998) Nonlinear component analysis as a kernel eigenvalue problem. Neural Comput 10(5): 1299–1319CrossRefGoogle Scholar
  12. Smola A, Kondor R (2003) Kernels and regularization on graphs. In: Learning theory and kernel machines: 16th annual conference on learning theory and 7th kernel workshop, COLT/Kernel 2003, Washington, August 24–27, 2003, proceedings. Springer, Berlin, p 144Google Scholar
  13. Tenenbaum J, Silva V, Langford J (2000) A global geometric framework for nonlinear dimensionality reduction. Science 290(5500): 2319CrossRefGoogle Scholar
  14. Zhou D, Schölkopf B (2005) Regularization on discrete spaces. Pattern Recognit 361: 361–368CrossRefGoogle Scholar

Copyright information

© The Author(s) 2012

Authors and Affiliations

  1. 1.Industrial EngineeringHanyang UniversitySeoulRepublic of Korea
  2. 2.College of ComputingGeorgia Institute of TechnologyAtlantaUSA
  3. 3.Industrial and Systems EngineeringGeorgia Institute of TechnologyAtlantaUSA

Personalised recommendations