D19 - approaches have the disadvantage that since ∊ = 0,...

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6.3 Discussion This chapter contains the core material of the book. It shows how the learning theory results of Chapter 4 can be used to avoid the difficulties of using linear functions in the high dimensional kernel-induced feature spaces of Chapter 3 . We have shown how the optimisation problems resulting from such an approach can be transformed into dual convex quadratic programmes for each of the approaches adopted for both classification and regression. In the regression case the loss function used only penalises errors greater than a threshold . Such a loss function typically leads to a sparse representation of the decision rule giving significant algorithmic and representational advantages. If, however, we set = 0 in the case of optimising the 2-norm of the margin slack vector, we recover the regressor output by a Gaussian process with corresponding covariance function, or equivalently the ridge regression function. These
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Unformatted text preview: approaches have the disadvantage that since ∊ = 0, the sparseness of the representation has been lost. The type of criterion that is optimised in all of the algorithms we have considered also arises in many other contexts, which all lead to a solution with a dual representation. We can express these criteria in the general form where L is a loss function, ||·|| 퓗 a regulariser and C is the regularisation parameter. If L is the square loss, this gives rise to regularisation networks of which Gaussian processes are a special case. For this type of problem the solution can always be expressed in the dual form. In the next chapter we will describe how these optimisation problems can be solved efficiently, frequently making use of the sparseness of the solution when deriving algorithms for very large datasets....
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This note was uploaded on 10/15/2011 for the course MBAHRM 565 taught by Professor Profbhattacharya during the Spring '11 term at IIT Kanpur.

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