Abstract
We study the class of hypothesis composed of linear functionals superimposed with smooth feature maps. We show that for “typical” smooth feature map, the pointwise convergence of hypothesis implies the convergence of some standard metrics such as error rate or area under ROC curve with probability 1 in selection of the test sample from a (Lebesgue measurable) probability density. Proofs use transversality theory. The crux is to show that for every “typical”, sufficiently smooth feature map into a finite dimensional vector space, the counter-image of every affine hyperplane has Lebesgue measure 0.
The results extend to every real analytic, in particular polynomial, feature map if its domain is connected and the limit hypothesis is non-constant. In the process we give an elementary proof of the fundamental lemma that locus of zeros of a real analytic function on a connected domain either fills the whole space or forms a subset of measure 0.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Platt, J.: Fast training of support vector machines using sequential minimal optimization. In: Schölkopf, B., Burges, C., Smola, A. (eds.) Advances in kernel methods: support vector learning, pp. 185–208. MIT Press, Cambridge, MA (1998)
Golubitsky, M., Guillemin, V.: Stable Mapping and Their Singularities. Springer, New York (1973)
Arnold, V., Gussein-Zade, S., Varchenko, A.: Singularities of Differentiable Maps. Birkhauser, Boston (1985)
Demazure, M.: Bifurcations and Catastrophes. Springer, New York (2000)
Vapnik, V.: Statistical Learning Theory. John Wiley and Sons, New York (1998)
Cristianini, N., Shawe-Taylor, J.: An Introduction to Support Vector Machines. Cambridge University Press, Cambridge, UK (2000)
Schölkopf, B., Smola, A.: Learning with Kernels. MIT Press, Cambridge, MA (2002)
Bartle, R.: The Elements of Integration and Lebesgue Measure. Wiley, Chichester (1995)
Provost, F., Fawcett, T.: Robust classification for imprecise environments. Machine Learning 42(3), 203–231 (2001)
Bamber, D.: The area above the ordinal dominance graph and the area below the receiver operating characteristic graph. J. Math. Psych. 12, 387–415 (1975)
Albert, A.: Regression and the Moore-Penrose Pseudoinverse. Academic Press, New York (1972)
Bedo, J., Sanderson, C., Kowalczyk, A.: An efficient alternative to svm based recursive feature elimination with applications in natural language processing and bioinformatics. In: Sattar, A., Kang, B.-H. (eds.) AI 2006. LNCS (LNAI), vol. 4304, pp. 170–180. Springer, Heidelberg (2006)
Kuratowski, K.: Introduction to Set Theory and Topology. PWN, Warszawa (1962)
Sternberg, S.: Lectures on Differential Geometry. Prentice-Hall, N.J (1964)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2007 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Kowalczyk, A. (2007). Continuity of Performance Metrics for Thin Feature Maps. In: Hutter, M., Servedio, R.A., Takimoto, E. (eds) Algorithmic Learning Theory. ALT 2007. Lecture Notes in Computer Science(), vol 4754. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-75225-7_28
Download citation
DOI: https://doi.org/10.1007/978-3-540-75225-7_28
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-75224-0
Online ISBN: 978-3-540-75225-7
eBook Packages: Computer ScienceComputer Science (R0)