Abstract
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.
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References
Belkin M, Niyogi P (2003) Laplacian eigenmaps for dimensionality reduction and data representation. Neural Comput 15(6): 1373–1396
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 processing
Donoho DL, Grimes C (2003) Hessian eigenmaps: locally linear embedding techniques for high-dimensional data. Proc Natl Acad Sci USA 100(10): 5591–5596
Haykin S (2008) Neural networks: a comprehensive foundation. Prentice Hall, Upper Saddle River
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. http://doi.ieeecomputersociety.org/10.1109/TPAMI.2006.223
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: 327–342
Mahadevan S, Maggioni M (2006) Value function approximation with diffusion wavelets and Laplacian eigenfunctions. Adv Neural Inf Process Syst 18: 843
Mari D, Kotz S (2001) Correlation and dependence. Imperial College Press, London
Nelsen R (2006) An introduction to copulas. Springer, New York
Roweis ST, Saul LK (2000) Nonlinear dimensionality reduction by locally linear embedding. Science 290(5500): 2323–2326. doi:10.1126/science.290.5500.2323
Scholkopf B, Smola A, Muller K (1998) Nonlinear component analysis as a kernel eigenvalue problem. Neural Comput 10(5): 1299–1319
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 144
Tenenbaum J, Silva V, Langford J (2000) A global geometric framework for nonlinear dimensionality reduction. Science 290(5500): 2319
Zhou D, Schölkopf B (2005) Regularization on discrete spaces. Pattern Recognit 361: 361–368
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Lee, K., Gray, A. & Kim, H. Dependence maps, a dimensionality reduction with dependence distance for high-dimensional data. Data Min Knowl Disc 26, 512–532 (2013). https://doi.org/10.1007/s10618-012-0267-9
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DOI: https://doi.org/10.1007/s10618-012-0267-9