On the Eigenspectrum of the Gram Matrix and its Relationship to the Operator Eigenspectrum (invited lecture for ALT 2002 and DS 2002) Authors: John Shawe-Taylor1, Chris Williams2, Nello Cristianini3, and Jaz Kandola1 Affiliation1: Department of Computer Science, Royal Holloway, University of London, Egham, Surrey, UK Affiliation2:University of Edinburgh Affiliation3:University of California at Davies Abstract. In this paper we analyze the relationships between the eigenvalues of the m x m Gram matrix K for a kernel k(· , ·) corresponding to a sample x1, . . . , xm drawn from a density p(x) and the eigenvalues of the corresponding continuous eigenproblem. We bound the differences between the two spectra and provide a performance bound on kernel PCA. ©Copyright 2002 Author and Springer