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By Johan A. K. Suykens

ISBN-10: 1586033417

ISBN-13: 9781586033415

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Developing a theory that allows us to create kernels that possess desirable properties (for example that can enforce desirable invariants). 4. Developing a new type of induction inference that is based on direct generalization from the training set to the test set, avoiding the intermediate problem of estimating a function (the transductive type inference). The hope is that this very fast growing area of research will significantly boost all branches of data analysis. This page intentionally left blank 27 Bibliography [1] N.

1-12) Let us characterize the diversity of this set of functions Q(z, a], a e A on the given sample by a quantity NA(ZI, . . , z g ) that represents the number of different separations of this sample that can be obtained using functions from the given set of indicator functions. Let us write this in another form. ,Q(zt,a))t a 6A that one obtains when a takes various values from A. Then geometrically speaking N*(ZI, . . , zt] is the number of different vertices of the ^-dimensional cube that can be obtained on the basis of the sample z\, .

For many different probability measures). The next question is then: Under what conditions is the ERM principle consistent and rapidly converging independently of the probability measure? The following equation describes the necessary and sufficient conditions for consistency of ERM for any probability measure: ,. n lim — -- = 0. 1-+00 i This condition is also sufficient for fast convergence. This equation is the third milestone in Statistical Learning Theory. It describes the conditions under which the learning machine implementing the ERM principle has an asymptotic high rate of convergence, independent of the problem to be solved.

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