![]() Firstly, a large number of support vectors (relative to the total amount of training data) should immediately raise eyebrows as it suggests overfitting. These are the support vectors that give the approach its name. Stepping back for a moment from the technical detail, intuitively what is happening here is that the algorithm is learning which class exemplars are the “most problematic” ones, i.e., which exemplars are nearest to the class boundaries and thus most likely to be misclassified. Support vector–based approaches usually perform well even with relatively small training datasets and have the advantage of well-understood mathematical behavior (which is an important consideration in the context of regularly compliance, among others). The regularizing parameter λ penalizes prediction errors. Subject to the constraints ∑ i = 1 n c i y i = 0 and 0 ≤ c i ≤ 1 / ( 2 n λ ). However, the computational complexity can be reduced, and learning accelerated, if special-purpose algorithms for training support vector machines are applied-but the details of these algorithms lie beyond the scope of this book (see Platt, 1998).Ĩ.7 ∑ i = 1 n c i − 1 2 ∑ i = 1 n ∑ j = 1 n y i c i k ( x i, x j ) y j c j There are off-the-shelf software packages for solving these problems (see Fletcher, 1987, for a comprehensive and practical account of solution methods). It turns out that finding the support vectors for the training instances and determining the parameters b and α i belongs to a standard class of optimization problems known as constrained quadratic optimization. Finally, b and α i are parameters that determine the hyperplane, just as the weights w 0, w 1, and w 2 are parameters that determine the hyperplane in the earlier formulation. If you are not familiar with dot product notation, you should still be able to understand the gist of what follows: Just think of a(i) as the whole set of attribute values for the ith support vector.
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