Abstract
Discriminative dimensionality reduction aims at a low dimensional, usually nonlinear representation of given data such that information as specified by auxiliary discriminative labeling is presented as accurately as possible. This paper centers around two open problems connected to this question: (i) how to evaluate discriminative dimensionality reduction quantitatively? (ii) how to arrive at explicit nonlinear discriminative dimensionality reduction mappings? Based on recent work for the unsupervised case, we propose an evaluation measure and an explicit discriminative dimensionality reduction mapping using the Fisher information.
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References
Baudat, G., Anouar, F.: Generalized discriminant analysis using a kernel approach. Neural Computation 12, 2385–2404 (2000)
Bunte, K., Biehl, M., Hammer, B.: A general framework for dimensionality reducing data visualization mapping. Neural Computation 24(3), 771–804 (2012)
Bunte, K., Schneider, P., Hammer, B., Schleif, F.-M., Villmann, T., Biehl, M.: Limited rank matrix learning, discriminative dimension reduction and visualization. Neural Networks 26, 159–173 (2012)
Cohn, D.: Informed projections. In: Becker, S., Thrun, S., Obermayer, K. (eds.) NIPS, pp. 849–856. MIT Press (2003)
Geng, X., Zhan, D.-C., Zhou, Z.-H.: Supervised nonlinear dimensionality reduction for visualization and classification. IEEE Transactions on Systems, Man, and Cybernetics, Part B 35(6), 1098–1107 (2005)
Gisbrecht, A., Lueks, W., Mokbel, B., Hammer, B.: Out-of-sample kernel extensions for nonparametric dimensionality reduction. In: ESANN 2012, pp. 531–536 (2012)
Gisbrecht, A., Mokbel, B., Hammer, B.: Linear basis-function t-sne for fast nonlinear dimensionality reduction. In: IJCNN (2013)
Goldberger, J., Roweis, S., Hinton, G., Salakhutdinov, R.: Neighbourhood components analysis. In: Advances in Neural Information Processing Systems 17, pp. 513–520. MIT Press (2004)
Hastie, T., Tibshirani, R., Friedman, J.: The Elements of Statistical Learning. Springer Series in Statistics. Springer New York Inc., New York (2001)
Iwata, T., Saito, K., Ueda, N., Stromsten, S., Griffiths, T.L., Tenenbaum, J.B.: Parametric embedding for class visualization. Neural Computation 19(9), 2536–2556 (2007)
Kaski, S., Sinkkonen, J., Peltonen, J.: Bankruptcy analysis with self-organizing maps in learning metrics. IEEE Transactions on Neural Networks 12, 936–947 (2001)
LeCun, Y., Bottou, L., Bengio, Y., Haffner, P.: Gradient-based learning applied to document recognition. Proceedings of the IEEE 86, 2278–2324 (1998)
Lee, J., Verleysen, M.: Quality assessment of dimensionality reduction: Rank-based criteria. Neurocomputing 72(7-9), 1431–1443 (2009)
Lee, J.A., Verleysen, M.: Nonlinear dimensionality redcution. Springer (2007)
Lee, J.A., Verleysen, M.: Scale-independent quality criteria for dimensionality reduction. Pattern Recognition Letters 31, 2248–2257 (2010)
Ma, B., Qu, H., Wong, H.: Kernel clustering-based discriminant analysis. Pattern Recognition 40(1), 324–327 (2007)
Memisevic, R., Hinton, G.: Multiple relational embedding. In: Saul, L.K., Weiss, Y., Bottou, L. (eds.) Advances in Neural Information Processing Systems 17, pp. 913–920. MIT Press, Cambridge (2005)
Peltonen, J., Klami, A., Kaski, S.: Improved learning of riemannian metrics for exploratory analysis. Neural Networks 17, 1087–1100 (2004)
van der Maaten, L.J.P., Hinton, G.E.: Visualizing high-dimensional data using t-sne. Journal of Machine Learning Research 9, 2579–2605 (2008)
Venna, J.: Dimensionality reduction for Visual Exploration of Similarity Structures. PhD thesis, Helsinki University of Technology, Espoo, Finland (2007)
Venna, J., Peltonen, J., Nybo, K., Aidos, H., Kaski, S.: Information retrieval perspective to nonlinear dimensionality reduction for data visualization. Journal of Machine Learning Research 11, 451–490 (2010)
Witten, D.M., Tibshirani, R.: Supervised multidimensional scaling for visualization, classification, and bipartite ranking. Computational Statistics and Data Analysis 55, 789–801 (2011)
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Gisbrecht, A., Hofmann, D., Hammer, B. (2012). Discriminative Dimensionality Reduction Mappings. In: Hollmén, J., Klawonn, F., Tucker, A. (eds) Advances in Intelligent Data Analysis XI. IDA 2012. Lecture Notes in Computer Science, vol 7619. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-34156-4_13
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DOI: https://doi.org/10.1007/978-3-642-34156-4_13
Publisher Name: Springer, Berlin, Heidelberg
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