Johan A. K. Suykens's Advances in learning theory: methods, models, and PDF

By Johan A. K. Suykens

ISBN-10: 1586033417

ISBN-13: 9781586033415

ISBN-10: 427490587X

ISBN-13: 9784274905872

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Example. Consider a mapping that allows us to construct decision polynomials in the input space. To construct a polynomial of degree two, one can create a feature space, Z, which has N = n(n+3> coordinates of the form: z\ = Xi, . . , zn = xn , n coordinates , 22 V. Vapnik zn+i = Zi, . . , 22n = #n , n coordinates , • • ZN = xnxn-i , 2i±iJ coordinates where x = (x^, . . , xn}. The separating hyperplane constructed in this space is a separating second degree polynomial in the input space. To construct a polynomial of degree A; in an n dimensional input space one has to construct a O(nk] dimensional feature space, where one then constructs the optimal hyperplane.

Note that both the equation describing the necessary and sufficient condition for consistency and the one that describes the sufficient condition for fast convergence of the ERM method are valid for a given probability measure P(z) (both VC-entropy HA(£) and VC-annealed entropy H^nn(f) are constructed using this measure). e. 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?

I=l 37 Best Choices for Regularization Parameters in Learning Theory Note that, since max{Mp, ||/p||oo + x/Ctf^y} < M -f 7 the confidence above is at least Applying this to £ = x\,... ,xm and writing the m resulting inequalities in matrix form we obtain that, with confidence at least the one in the statement, 1 1 rrry <2e. D Lemma 6 For all 7, e > 0, 7m / " PROOF. , xm and writing the resulting m equalities in matrix form we obtain 7"^/7,z[x] + /^ztxjATfx] = K[x]y from which the statement follows.

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Advances in learning theory: methods, models, and applications by Johan A. K. Suykens


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