jmlr jmlr2007 jmlr2007-46 jmlr2007-46-reference knowledge-graph by maker-knowledge-mining
Source: pdf
Author: Marco Reisert, Hans Burkhardt
Abstract: This paper presents a new class of matrix valued kernels that are ideally suited to learn vector valued equivariant functions. Matrix valued kernels are a natural generalization of the common notion of a kernel. We set the theoretical foundations of so called equivariant matrix valued kernels. We work out several properties of equivariant kernels, we give an interpretation of their behavior and show relations to scalar kernels. The notion of (ir)reducibility of group representations is transferred into the framework of matrix valued kernels. At the end to two exemplary applications are demonstrated. We design a non-linear rotation and translation equivariant filter for 2D-images and propose an invariant object detector based on the generalized Hough transform. Keywords: kernel methods, matrix kernels, equivariance, group integration, representation theory, Hough transform, signal processing, Volterra series
L. Amodei. Reproducing kernels of vector-valued function spaces. In Surface Fitting and Multiresolution Methods, A. Le Maut C. Rabut and L. L. Schumaker (eds.), pages 17–26, 1996. N. Aronszajn. Theory of reproducing kernels. Trans. AMS, 686:337404, 1950. D.H. Ballard. Generalizing the hough transform to detect arbitrary shapes. Pattern Recognition, 13-2, 1981. S. Boyd, L. O. Chua, and C. A. Desoer. Analytical foundations of Volterra series. IMA Journal of Mathematical Control and Information, 1:243–282, 1984. J. Burbea and P. Masani. Banach and Hilbert spaces of vector-valued functions. Pitman Research Notes in Mathematics, 90, 1984. H. Burkhardt and S. Siggelkow. Invariant features in pattern recognition - fundamentals and applications. In In Nonlinear Model-Based Image/Video Processing and Analysis, pages 269–307. John Wiley and Sons, 2001. N. Canterakis. Vollstaendige minimale systeme von polynominvarianten fuer die zyklischen gruppen g(n) und fuer g(n) x g(n). In In Tagungsband des 9. Kolloquiums - DFG Schwerpunkt: Digitale Signalverarbeitung, pages 13–17, 1986. T.J. Dodd and R.F. Harrison. Estimating Volterra filters in Hilbert space. In Proceedings of IFAC Conference on Intelligent Control Systems and Signal Processing, 2003. A. Gaal. Linear Analysis and Represenatation Theory. Springer Verlag, New York, 1973. B. Haasdonk, A. Vossen, and H. Burkhardt. Invariance in kernel methods by haar-integration kernels. In Proceedings of the 14th Scandinavian Conference on Image Analysis, pages 841–851, 2005. A.K. Jain. Fundamentals of Digital Image Processing. Prentice-Hall, Inc., Englewood Cliffs, N.J., USA, 1989. G.S. Kimeldorf and G. Wahba. Some results on Tchebycheffian spline functions. Journal of Mathematical Analysis and Applications, 33:82–95, 1971. R. Lenz. Group Theoretical Methods in Image Processing. Springer Verlag, Lecture Notes, 1990. D.G. Lowe. Distinct image features from scale-invariant keypoints. International Journal of Computer Vision, 60:91–110, 2004. C.A. Micchelli and M. Pontil. On learning vector-valued functions. Neural Computation, 17:177– 204, 2005. W. Miller. Topics in harmonic analysis with applications to radar and sonar. IMA Volumes in Mathematics and its Applications, 1991. D. Mumford, J. Fogarty, and F. Kirwan. Geometric Invariant Theory. Springer, 1994. 407 R EISERT AND B URKHARDT L. Nachbin. The Haar Integral. D. van Nostrand Company, Inc., Princenton, New Jersey, Toronto, New York, London, 1965. M. Pontil and C.A. Micchelli. Kernels for multi-task learning. In Proceeding of the NIPS, 2004. V. I. Paulsen R. G. Douglas. Hilbert Modules over Function Algebras. Wiley New York, 1989. M. Reisert and H. Burkhardt. Averaging similarity weighted group representations for pose estimation. In Proceedings of IVCNZ’05, 2005. B. Schoelkopf and A. J. Smola. Learning with Kernels. The MIT Press, 2002. I. Schur. Vorlesungen ueber Invariantentheorie. Springer Verlag, Berlin, Heidelberg, New York, 1968. J. Shawe-Taylor and N. Cristianini. Kernel Methods for Pattern Analysis. Cambridge University Press, 2004. 408