Use support vector machines to solve XOR classification problems

Support vector machine is a commonly used classification algorithm suitable for linear and non-linear classification problems. This article will introduce how to use support vector machines to solve the XOR problem.

The XOR problem means that when the input contains two binary variables, the output is true (1) if the two variables are not equal, otherwise the output is false (0). For example, when the input is (0, 1) or (1, 0), the output is 1, and when the input is (0, 0) or (1, 1), the output is 0. This is a nonlinear problem because the two outputs cannot be separated using a single straight line.

To solve the XOR problem, support vector machines can achieve linear separability by mapping the input into a high-dimensional space. For example, we can map the input (x1, x2) to (x1, x2, x1 x2) in three-dimensional space. In this new space we can use a plane to separate the two outputs. We can then map the equations of this plane back to the original 2D space to obtain a decision boundary for classification. This can effectively solve the XOR problem.

Specifically, the kernel technique of support vector machine can be used to achieve this mapping. The kernel trick is a method of mapping an input into a high-dimensional space without having to explicitly compute this mapping. Commonly used kernel functions include linear kernel function, polynomial kernel function and radial basis function kernel function. In this example, we will use the RBF kernel function.

The following is the code for using Python to implement support vector machine to solve the XOR problem:

from sklearn import svm

# 输入数据
X = [[0, 0], [0, 1], [1, 0], [1, 1]]
# 输出数据
y = [0, 1, 1, 0]

# 定义SVM模型，使用RBF核函数
clf = svm.SVC(kernel='rbf')

#使用输入和输出数据训练模型
clf.fit(X, y)

# 预测新的输入数据
print(clf.predict([[0, 1], [1, 1], [0, 0], [1, 0]]))

In the code, we define an input data set X and an output data set y, and then train a support vector machine model on these data. We use the RBF kernel function to initialize the support vector machine model, and then call the fit() method to train the model. Finally, we use the predict() method to predict the new input data and print out the prediction results.

In this example, we used four input data points ([0,0], [0,1], [1,0], [1,1]) and the corresponding output data (0, 1, 1, 0). We map these points into three-dimensional space and separate them using the RBF kernel function. Finally, we have a classifier that can predict the output for new input data.

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