support vector machines编辑历史

# Support vector machine

A support vector machine (SVM) is a supervised machine learning model used for classification, regression, and outlier detection. It works by finding a hyperplane that best separates data into classes in a high-dimensional space, maximizing the margin between the closest points (support vectors) of each class. SVMs are particularly effective for binary classification and can handle non‑linear decision boundaries through the kernel trick, making them a fundamental tool in information technology and statistical learning theory.

## 1 Mathematical formulation

The mathematical foundation of SVMs rests on constructing an optimal separating hyperplane that maximizes the margin between classes, often in a transformed feature space.

### 1.1 Linear SVM

In the linear case, the decision boundary is a hyperplane defined by \(\mathbf{w} \cdot \mathbf{x} + b = 0\), where \(\mathbf{w}\) is the normal vector and \(b\) is the bias. The goal is to maximize the margin, the distance between the hyperplane and the nearest points of either class.

#### 1.1.1 Hard‑margin classification

Hard‑margin SVM assumes the data are perfectly linearly separable. The constraints are \(y_i (\mathbf{w} \cdot \mathbf{x}_i + b) \ge 1\) for all \(i\), where \(y_i \in \{-1, +1\}\). The margin is \(\frac{2}{\|\mathbf{w}\|}\), so maximizing it is equivalent to minimizing \(\frac{1}{2}\|\mathbf{w}\|^2\) subject to those constraints. This is a convex quadratic optimization problem with a unique global minimum.

#### 1.1.2 Soft‑margin classification (slack variables)

Real‑world data are rarely perfectly separable. Soft‑margin SVM introduces slack variables \(\xi_i \ge 0\) that allow some points to lie within the margin or on the wrong side. The objective becomes \(\frac{1}{2}\|\mathbf{w}\|^2 + C \sum_i \xi_i\), with constraints \(y_i (\mathbf{w} \cdot \mathbf{x}_i + b) \ge 1 - \xi_i\). The parameter \(C\) controls the trade‑off between margin width and misclassificat