Abstract
We revisit the kernel minimum enclosing ball problem and show that it can be solved using simple recurrent neural networks. Once solved, the interior of a ball can be characterized in terms of a function of a set of support vectors and local minima of this function can be thought of as prototypes of the data at hand. For Gaussian kernels, these minima can be naturally found via a mean shift procedure and thus via another recurrent neurocomputing process. Practical results demonstrate that prototypes found this way are descriptive, meaningful, and interpretable.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Bauckhage, C.: A neural network implementation of Frank-Wolfe optimization. In: Lintas, A., Rovetta, S., Verschure, P.F.M.J., Villa, A.E.P. (eds.) ICANN 2017. LNCS, vol. 10613, pp. 219–226. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-68600-4_26
Bauckhage, C., Thurau, C.: Making archetypal analysis practical. In: Denzler, J., Notni, G., Süße, H. (eds.) DAGM 2009. LNCS, vol. 5748, pp. 272–281. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-03798-6_28
Ben-Hur, A., Horn, D., Siegelmann, H., Vapnik, V.: Support vector clustering. J. Mach. Learn. Res. 2(Dec), 125–137 (2001)
Cheng, Y.: Mean shift, mode seeking, and clustering. IEEE Trans. Pattern Anal. Mach. Intell. 17(8), 767–776 (1995). https://doi.org/10.1109/34.400568
Cutler, A., Breiman, L.: Archetypal analysis. Technometrics 36(4), 338–347 (1994). https://doi.org/10.1080/00401706.1994.10485840
Dong, T., et al.: Imposing category trees onto word-embeddings using a geometric construction. In: Proceedings ICLR (2019)
Dong, T., Wang, Z., Li, J., Bauckhage, C., Cremers, A.: Triple classification using regions and fine-grained entity typing. In: Proceedings AAAI (2019)
Evangelista, P.F., Embrechts, M.J., Szymanski, B.K.: Some properties of the Gaussian kernel for one class learning. In: de Sá, J.M., Alexandre, L.A., Duch, W., Mandic, D. (eds.) ICANN 2007. LNCS, vol. 4668, pp. 269–278. Springer, Heidelberg (2007). https://doi.org/10.1007/978-3-540-74690-4_28
Frank, M., Wolfe, P.: An algorithm for quadratic programming. Naval Res. Logist. 3(1–2), 95–110 (1956)
Fukunaga, K., Hostetler, L.: The estimation of the gradient of a density function with applications in pattern recognition. IEEE Trans. Inf. Theory 21(1), 32–40 (1975). https://doi.org/10.1109/TIT.1975.1055330
Jäger, H., Haas, H.: Harnessing nonlinearity: predicting chaotic systems and saving energy in wireless communication. Science 304(5667), 78–80 (2004). https://doi.org/10.1126/science.1091277
LeCun, Y., Boottou, L., Bengio, Y., Haffner, P.: Gradient-based learning applied to document recognition. Proc. IEEE 86(11), 2278–2324 (1998). https://doi.org/10.1109/5.726791
Mahoney, M., Drineas, P.: CUR matrix decompositions for improved data analysis. PNAS 106(3), 697–702 (2009). https://doi.org/10.1073/pnas.0803205106
Ruff, L., et al.: Deep one-class classification. In: Proceedings ICML (2018)
Schleif, F.M., Gisbrecht, A., Tino, P.: Supervised low rank indefinite kernel approximation using minimum enclosing balls. Neurocomputing 318(Nov), 213–226 (2018). https://doi.org/10.1016/j.neucom.2018.08.057
Sifa, R.: An overview of Frank-Wolfe optimization for stochasticity constrained interpretable matrix and tensor factorization. In: Kůrková, V., Manolopoulos, Y., Hammer, B., Iliadis, L., Maglogiannis, I. (eds.) ICANN 2018. LNCS, vol. 11140, pp. 369–379. Springer, Cham (2018). https://doi.org/10.1007/978-3-030-01421-6_36
Tax, D., Duin, R.: Support vector data description. Mach. Learn. 54(1), 45–46 (2004). https://doi.org/10.1023/B:MACH.0000008084.60811.49
Thurau, C., Kersting, K., Bauckhage, C.: Deterministic CUR for improved large-scale data analysis: an empirical study. In: Proceedings SDM. SIAM (2012)
Thurau, C., Kersting, K., Wahabzada, M., Bauckhage, C.: Descriptive matrix factorization for sustainability: adopting the principle of opposites. Data Min. Knowl. Discov. 24(2), 325–354 (2012). https://doi.org/10.1007/s10618-011-0216-z
Tsang, I., Kwok, J., Cheung, P.M.: Core vector machines: fast SVM training on very large data sets. J. Mach. Learn. Res. 6(Apr), 363–392 (2010)
Wang, S., Zhang, Z.: Improving CUR matrix decompositions and the Nyström approximation via adaptive sampling. J. Mach. Learn. Res. 14(1), 2729–2769 (2010)
Xiao, H., Rasul, K., Vollgraf, R.: Fashion-MNIST: a novel image dataset for benshmarking machine learning algorithms. arXiv:1708.07747 [cs.LG] (2017)
Zhang, K., Kwok, J.: Clustered Nyström method for large scale manifold learning and dimension reduction. IEEE Trans. Neural Netw. 21(10), 1576–1587 (2010). https://doi.org/10.1109/TNN.2010.2064786
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this paper
Cite this paper
Bauckhage, C., Sifa, R., Dong, T. (2019). Prototypes Within Minimum Enclosing Balls. In: Tetko, I., Kůrková, V., Karpov, P., Theis, F. (eds) Artificial Neural Networks and Machine Learning – ICANN 2019: Workshop and Special Sessions. ICANN 2019. Lecture Notes in Computer Science(), vol 11731. Springer, Cham. https://doi.org/10.1007/978-3-030-30493-5_36
Download citation
DOI: https://doi.org/10.1007/978-3-030-30493-5_36
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-30492-8
Online ISBN: 978-3-030-30493-5
eBook Packages: Computer ScienceComputer Science (R0)