SupportVectorMaschinen

SupportVectorMaschinen, commonly known in English as support vector machines (SVM), are supervised learning models used for classification and regression tasks. They aim to find a decision boundary that separates data points of different classes with the largest possible margin in the feature space. The boundary is defined by a subset of the training samples called support vectors, which determine the position and orientation of the separating hyperplane.

In the linear case, a central objective is to maximize the margin between classes while correctly classifying training examples. This leads to a convex optimization problem that minimizes the norm of the weight vector subject to correct classification constraints. In practice, many problems are not perfectly separable, so a soft margin is used with slack variables and a regularization parameter C that trades off margin size against misclassification. The problem has a dual formulation that relies on Lagrange multipliers, and it can be expressed with kernel functions to operate in higher-dimensional feature spaces without explicit mapping.

The kernel trick enables SVMs to handle nonlinear decision boundaries. By choosing a kernel function K(xi, xj) that corresponds to an implicit feature map, the algorithm computes inner products in a high-dimensional space efficiently. Common kernels include linear, polynomial, and radial basis function (RBF, Gaussian) kernels, as well as sigmoid. Variants such as ν-SVM adjust the margin control, and support vector regression (SVR) extends the approach to continuous targets. For multi-class problems, strategies like one-vs-one and one-vs-rest are typically employed.

Applications of SupportVectorMaschinen span text classification, image recognition, bioinformatics, and anomaly detection. They perform well in high-dimensional spaces and when the number of features is large relative to the number of samples, but training can be computationally intensive and sensitive to kernel choice and parameter tuning.