Kernel ridge regression C++实现

以下是 Kernel ridge regression 的 C++ 实现示例：

#include <iostream>
#include <vector>
#include <cmath>

using namespace std;

// 高斯核函数
double gaussian_kernel(double x, double y, double sigma) {
    double diff = x - y;
    return exp(-diff * diff / (2 * sigma * sigma));
}

// 计算带核函数的 Gram 矩阵
vector<vector<double>> compute_gram_matrix(const vector<double>& x, double sigma) {
    int n = x.size();
    vector<vector<double>> gram_matrix(n, vector<double>(n));
    for (int i = 0; i < n; i++) {
        for (int j = 0; j < n; j++) {
            gram_matrix[i][j] = gaussian_kernel(x[i], x[j], sigma);
        }
    }
    return gram_matrix;
}

// 计算 alpha
vector<double> compute_alpha(const vector<vector<double>>& gram_matrix, const vector<double>& y, double alpha) {
    int n = gram_matrix.size();
    vector<vector<double>> I(n, vector<double>(n));
    for (int i = 0; i < n; i++) {
        I[i][i] = 1;
    }

    vector<vector<double>> temp(n, vector<double>(n));
    for (int i = 0; i < n; i++) {
        for (int j = 0; j < n; j++) {
            temp[i][j] = gram_matrix[i][j] + alpha * I[i][j];
        }
    }

    // 求解线性方程组
    vector<double> alpha_vec(n);
    for (int i = 0; i < n; i++) {
        double sum = 0;
        for (int j = 0; j < n; j++) {
            sum += temp[i][j] * y[j];
        }
        alpha_vec[i] = sum;
    }

    return alpha_vec;
}

// 预测
double predict(const vector<vector<double>>& gram_matrix, const vector<double>& alpha, double x, double sigma) {
    double sum = 0;
    int n = gram_matrix.size();
    for (int i = 0; i < n; i++) {
        sum += alpha[i] * gaussian_kernel(x, i, sigma);
    }
    return sum;
}

int main() {
    // 构造数据集
    vector<double> x = {1, 2, 3, 4, 5};
    vector<double> y = {2, 3, 1, 5, 7};

    // 计算带核函数的 Gram 矩阵
    double sigma = 1;
    vector<vector<double>> gram_matrix = compute_gram_matrix(x, sigma);

    // 计算 alpha
    double alpha = 0.1;
    vector<double> alpha_vec = compute_alpha(gram_matrix, y, alpha);

    // 预测
    double x_test = 6;
    double y_test = predict(gram_matrix, alpha_vec, x_test, sigma);

    cout << "Predicted value: " << y_test << endl;

    return 0;
}

以上代码实现了基于高斯核函数的 Kernel ridge regression，通过计算带核函数的 Gram 矩阵和求解线性方程组来确定 alpha，然后使用 alpha 进行预测。需要注意的是，Kernel ridge regression 的核函数可以是任意合法的核函数，这里只是提供了高斯核函数的实现。