Making Archetypal Analysis Practical
Conference paper
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Abstract
Archetypal analysis represents the members of a set of multivariate data as a convex combination of extremal points of the data. It allows for dimensionality reduction and clustering and is particularly useful whenever the data are superpositions of basic entities. However, since its computation costs grow quadratically with the number of data points, the original algorithm hardly applies to modern pattern recognition or data mining settings. In this paper, we introduce ways of notably accelerating archetypal analysis. Our experiments are the first successful application of the technique to large scale data analysis problems.
Keywords
Convex Hull Convex Combination Natural Image Nonnegative Matrix Factorization Nonnegative Matrix
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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References
- 1.Cutler, A., Breiman, L.: Archetypal Analysis. Technometrics 36(4), 338–347 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
- 2.Jolliffe, I.: Principal Component Analysis. Springer, Heidelberg (1986)CrossRefzbMATHGoogle Scholar
- 3.Schölkopf, B., Smola, A.J., Müller, K.-R.: Nonlinear Component Analysis as a Kernel Eigenvalue Problem. Neural Computation 10(5), 1299–1319 (1998)CrossRefGoogle Scholar
- 4.Lee, D.D., Seung, S.: Learning the Parts of Objects by Non-Negative Matrix Factorization. Nature 401(6755), 788 (1999)CrossRefGoogle Scholar
- 5.Finesso, L., Spreij, P.: Approximate Nonnegative Matrix Factorization via Alternating Minimization. In: Proc. 16th Int. Symp. on Mathematical Theory of Networks and Systems, Leuven (July 2004)Google Scholar
- 6.Stone, E., Cutler, A.: Archetypal Analysis of Spatio-temporal Dynamics. Physica D 90(3), 209–224 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
- 7.Chan, B.H.P.: Archetypal Analysis of Galaxy Spectra. Monthly Notices of the Royal Astronomical Society 338(3), 790–795 (2003)CrossRefGoogle Scholar
- 8.Huggins, P., Pachter, L., Sturmfels, B.: Toward the Human Genotope. Bulletin of Mathematical Biology 69(8), 2723–2735 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
- 9.Joachims, T.: Making Large-Scale Support Vector Machine Learningn Practical. In: Advances in Kernel Methods: Support Vector Learning, MIT Press, Cambridge (1999)Google Scholar
- 10.de Berg, M., van Kreveld, M., Overmars, M., Schwarzkopf, O.: Computational Geometry. Springer, Heidelberg (2000)CrossRefzbMATHGoogle Scholar
- 11.Ziegler, G.M.: Lectures on Polytopes. Springer, Heidelberg (1995)CrossRefzbMATHGoogle Scholar
- 12.Donoho, D.L., Tanner, J.: Neighborliness of Randomly-Projected Simplices in High Dimensions. Proc. of the Nat. Academy of Sciences 102(27), 9452–9457 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
- 13.Hall, P., Marron, J., Neeman, A.: Geometric representation of high dimension low sample size data. J. of the Royal Statistical Society B 67(3), 427–444 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
- 14.Blank, M., Gorelick, L., Shechtman, E., Irani, M., Basri, R.: Actions as Space-Time Shapes. In: Proc. ICCV (2005)Google Scholar
- 15.Torralba, A., Fergus, R., Freeman, W.T.: 80 Million Tiny Images: A Large Dataset for Non-parametric Object and Scene Recognition. IEEE Trans. on Pattern Analalysis and Machine Intelligence 30(11), 1958–1970 (2008)CrossRefGoogle Scholar
- 16.Heidemann, G.: The principal components of natural images revisited. IEEE Trans. on Pattern Analalysis and Machine Intelligence 28(5), 822–826 (2006)CrossRefGoogle Scholar
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