Which is the Lagrangian formulation of the SVM?
Having introduced some elements of statistical learning and demonstrated the potential of SVMs for company rating we can now give a Lagrangian formulation of an SVM for the linear classification problem and generalize this approach to a nonlinear case. Figure 10.3: The separating hyperplane and the margin in a non-separable case.
What is the idea of soft margin formulation?
Soft Margin Formulation. This idea is based on a simple premise: allow SVM to make a certain number of mistakes and keep margin as wide as possible so that other points can still be classified correctly. This can be done simply by modifying the objective of SVM.
When to use conditional minimization in the SVM?
The conditional minimization of the objective function with constraint ( 10.9) and ( 10.10) provides the highest possible margin in the case when classification errors are inevitable due to the linearity of the separating hyperplane. Under such a formulation the problem is convex. One can show that margin maximization reduces the VC dimension.
Can a SVM be generalized to the nonlinear case?
The SVMs can also be easily generalized to the nonlinear case. It is worth noting that all the training vectors appear in the dual Lagrangian formulation only as scalar products. This means that we can apply kernels to transform all the data into a high dimensional Hilbert feature space and use linear algorithms there:
What is the basic idea of the SVM?
In the linear case the following inequalities hold for all points of the training set: The basic idea of the SVM classification is to find such a separating hyperplane that corresponds to the largest possible margin between the points of different classes, see Figure 10.3. Some penalty for misclassification must also be introduced.
Is the constraint x = 1 useless in SVM?
Indeed, the constraint x = 1 is violated. In this example, our only choice is to use x = 1 and this is the solution. Looking at this example, you might feel like equality constraints are useless. This is not the case because most of the time optimization problems are performed in more than one dimension.