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Translation Invariance in the Polynomial Kernel Space and Its Applications in kNN Classification

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Abstract

In this paper, a new technique is presented to measure dissimilarity in kernel space providing scaling and translation invariance. The motivation comes from signal/image processing, where classifiers are often required to ensure invariance against linear transforms, since in many cases linear transforms do not affect the content of a signal/image for a human observer. We examine the theoretical background of linear invariance in the polynomial kernel space, introduce the centered correlation and centered Euclidean dissimilarity in kernel space, deduce formulas to compute it efficiently and test the proposed dissimilarity measures with the kNN classifier. The experimental results show that the presented techniques are highly competitive in similarity or dissimilarity based classification methods.

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References

  1. Blake CL, Merz CJ (1998) UCI repository of machine learning databases. University of California, Irvine

  2. Blazadonakis ME, Zervakis M (2007) Polynomial and RBF kernels as marker selection tools—a breast cancer case study. In: Fourth international conference on machine learning and applications, pp 488–493. http://doi.ieeecomputersociety.org/10.1109/ICMLA.2007.67

  3. Cai R, Hao Z, Wen W, Huang H (2010) Kernel based gene expression pattern discovery and its application on cancer classification. Neurocomputing 73:2562–2570. doi:10.1016/j.neucom.2010.05.019

    Google Scholar 

  4. Camastra F (2006) Kernel methods for clustering. In: WIRN/NAIS, Lecture Notes in Computer Science vol 3931, pp 1–9

  5. Daze MM, Daze E (2009) Encyclopedia of distances. Springer, New York

    Google Scholar 

  6. Denton A, Perrizo WA (2004) Kernel-based semi-naïve Bayesian classifier using p-trees. In: SDM’04, pp 427–431

  7. Evans M, Hastings N, Peacock B (2000) Statistical distributions. Wiley-Interscience. http://www.amazon.com/exec/obidos/redirect?tag=citeulike07-20&path=ASIN/0471371246

  8. Graepel T (1998) Self-organizing maps: generalizations and new optimization techniques. Neurocomputing 21(1–3):173–190. doi:10.1016/S0925-2312(98)00035-6

    Google Scholar 

  9. Herbrich R (2001) Learning Kernel classifiers: theory and algorithms (adaptive computation and machine learning). The MIT Press. http://www.amazon.com/exec/obidos/redirect?tag=citeulike07-20&path=ASIN/026208306X

  10. Hofmann T, Scholkopf B, Smola AJ (2008) Kernel methods in machine learning. http://arxiv.org/abs/math.ST/0701907

  11. Jadhav D, Kulkarni J, Holambe R (2008) Multiresolution feature based fractional power polynomial kernel fisher discriminant model for face recognition. J Multimedia 3(1). http://ojs.academypublisher.com/index.php/jmm/article/view/03014753

  12. Kulkarni SR, Lugosi G, Venkatesh SS (1998) Learning pattern classification—a survey. IEEE Trans Inf Theory 44:2178–2206

    Google Scholar 

  13. Lampert CH (2009) Kernel methods in computer vision. Found Trends Comput Graph Vis 4:193–285. doi:10.1561/0600000027. http://portal.acm.org/citation.cfm?id=1640474.1640475

    Google Scholar 

  14. Minh HQ (2006) Reproducing kernel Hilbert spaces in learning theory. Dissertation, Brown University

  15. Minh H, Niyogi P, Yao Y (2006) Mercers theorem, feature maps, and smoothing. In: Lugosi G, Simon H (eds) Learning theory. Lecture notes in computer science, vol 4005. Springer, Berlin, pp 154–168

  16. Nehe NS, Holambe RS (2009) Isolated word recognition using low dimensional features and kernel based classification. In: ARTCom, pp 194–198

  17. Niemeijer M, Van Ginneken B, Staal J, Suttorp-Schulten MSA, Abramoff MD (2005) Automatic detection of red lesions in digital color fundus photographs. IEEE Trans Med Imaging 24(5): 584–592

    Article  Google Scholar 

  18. Niemeijer M, van Ginneken B, Cree MJ, Mizutani A, Quellec G, Sanchez CI, Zhang B, Hornero R, Lamard M, Muramatsu C, Wu X, Cazuguel G, You J, Mayo A, Li Q, Hatanaka Y, Cochener B, Roux C, Karray F, Garcia M, Fujita H, Abramoff MD (2010) Retinopathy online challenge: automatic detection of microaneurysms in digital color fundus photographs. IEEE Trans Med Imaging 29(1):185–195. http://www.biomedsearch.com/nih/Retinopathy-Online-Challenge-Automatic-Detection/19822469.html

  19. Scholkopf B (2000) The Kernel trick for distances. In: NIPS, pp 301–307. http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.20.327

  20. Scholkopf B, Smola AJ (2001) Learning with Kernels: support vector machines, regularization, optimization, and beyond. MIT Press, Cambridge

    Google Scholar 

  21. Schlkopf B, Smola AJ, Müller KR (1999) Kernel principal component analysis. Advances in kernel methods: support vector learning, pp 327–352. http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.31.3580

  22. Takeda H, Milanfar P (2010) Nonlinear kernel backprojection for computed tomography. In: Proceedings of the IEEE international conference on acoustics, speech, and signal processing, ICASSP 2010, 14–19 March 2010, Sheraton Dallas Hotel, Dallas, TX, USA, pp 618–621. IEEE. doi:10.1109/ICASSP.2010.5495184

  23. Vapnik VN (1998) Statistical learning theory. Wiley-Interscience. http://www.amazon.com/exec/obidos/redirect?tag=citeulike07-20&path=ASIN/0471030031

  24. Yu K, Ji L, Zhang X (2002) Kernel nearest-neighbor algorithm. Neural Process Lett 15:147–156. doi:10.1023/A:1015244902967

    Google Scholar 

  25. Zhao G, Pietikzinen M (2006) Local binary pattern descriptors for dynamic texture recognition. In: Proceedings of the international conference on pattern recognition, pp 211–214

  26. Zhou J, Liu Y, Chen Y (2007) Face recognition using kernel PCA and hierarchical RBF network. In: Proceedings of the 6th international conference on computer information systems and industrial management applications. IEEE Computer Society, Washington, DC, pp 239–244. doi:10.1109/CISIM.2007.28. http://portal.acm.org/citation.cfm?id=1270379.1270523

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Authors and Affiliations

  1. Faculty of Informatics, University of Debrecen, POB 12, Debrecen, 4010, Hungary

    György Kovács & András Hajdu

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  1. György Kovács
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  2. András Hajdu
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Correspondence to György Kovács.

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Kovács, G., Hajdu, A. Translation Invariance in the Polynomial Kernel Space and Its Applications in kNN Classification. Neural Process Lett 37, 207–233 (2013). https://doi.org/10.1007/s11063-012-9242-0

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