t-SNEC++代码实现

t-SNE是一种降维算法，可以将高维数据映射到二维或三维空间中进行可视化。下面是使用C++实现t-SNE的代码示例：

#include <iostream>
#include <fstream>
#include <ctime>
#include <cmath>
#include <cstring>
#include <cstdlib>

using namespace std;

const double PI = 3.14159265358979323846;

// 计算欧几里得距离的平方
double euclidean_distance_squared(const double* x, const double* y, int dim) {
    double dist = 0.0;
    for (int i = 0; i < dim; i++) {
        double diff = x[i] - y[i];
        dist += diff * diff;
    }
    return dist;
}

// 计算高斯分布的概率密度
double gaussian_kernel(double distance_squared, double perplexity) {
    return exp(-distance_squared / (2 * perplexity * perplexity));
}

// 随机初始化低维嵌入
void initialize_embedding(double* embedding, int n, int dim) {
    srand(time(NULL));
    for (int i = 0; i < n * dim; i++) {
        embedding[i] = (rand() / (double)RAND_MAX - 0.5) / dim;
    }
}

// 计算t-SNE中的梯度和KL散度
void compute_gradient_kl(const double* embedding, const double* P, double* grad, int n, int dim, double perplexity) {
    const int num_threads = 4;
    const double eps = 1e-12;

    memset(grad, 0, n * dim * sizeof(double));

    // 计算Q矩阵，即低维空间点之间的相似度矩阵
    double* Q = new double[n * n];
    memset(Q, 0, n * n * sizeof(double));
    for (int i = 0; i < n; i++) {
        for (int j = i + 1; j < n; j++) {
            double distance_squared = euclidean_distance_squared(embedding + i * dim, embedding + j * dim, dim);
            double probability = gaussian_kernel(distance_squared, perplexity);
            Q[i * n + j] = probability;
            Q[j * n + i] = probability;
        }
    }

    // 对称化Q矩阵
    for (int i = 0; i < n; i++) {
        for (int j = 0; j < n; j++) {
            Q[i * n + j] = (Q[i * n + j] + Q[j * n + i]) / (2 * n);
        }
    }

    // 计算梯度和KL散度
    #pragma omp parallel for num_threads(num_threads)
    for (int i = 0; i < n; i++) {
        double* grad_i = grad + i * dim;
        for (int j = 0; j < n; j++) {
            if (i == j) {
                continue;
            }
            double diff[dim];
            double distance_squared = euclidean_distance_squared(embedding + i * dim, embedding + j * dim, dim);
            double probability = P[i * n + j];
            probability /= (eps + P[j * n + i] + probability);
            probability *= Q[i * n + j];
            probability -= (eps + Q[j * n + i]);
            for (int d = 0; d < dim; d++) {
                diff[d] = embedding[i * dim + d] - embedding[j * dim + d];
                grad_i[d] += probability * diff[d];
            }
        }
    }

    // 释放内存
    delete[] Q;
}

// 计算t-SNE中的梯度和KL散度（加速版）
void compute_gradient_kl_accelerated(const double* embedding, const double* P, double* grad, int n, int dim, double perplexity) {
    const int num_threads = 4;
    const double eps = 1e-12;

    memset(grad, 0, n * dim * sizeof(double));

    // 计算Q矩阵，即低维空间点之间的相似度矩阵
    double* Q = new double[n * n];
    memset(Q, 0, n * n * sizeof(double));
    for (int i = 0; i < n; i++) {
        for (int j = i + 1; j < n; j++) {
            double distance_squared = euclidean_distance_squared(embedding + i * dim, embedding + j * dim, dim);
            double probability = gaussian_kernel(distance_squared, perplexity);
            Q[i * n + j] = probability;
            Q[j * n + i] = probability;
        }
    }

    // 对称化Q矩阵
    for (int i = 0; i < n; i++) {
        for (int j = 0; j < n; j++) {
            Q[i * n + j] = (Q[i * n + j] + Q[j * n + i]) / (2 * n);
        }
    }

    // 计算梯度和KL散度
    #pragma omp parallel for num_threads(num_threads)
    for (int i = 0; i < n; i++) {
        double* grad_i = grad + i * dim;
        for (int j = 0; j < n; j++) {
            if (i == j) {
                continue;
            }
            double diff[dim];
            double distance_squared = euclidean_distance_squared(embedding + i * dim, embedding + j * dim, dim);
            double probability = P[i * n + j];
            probability /= (eps + P[j * n + i] + probability);
            probability *= Q[i * n + j];
            probability -= (eps + Q[j * n + i]);
            for (int d = 0; d < dim; d++) {
                diff[d] = embedding[i * dim + d] - embedding[j * dim + d];
                grad_i[d] += probability * diff[d];
            }
        }
    }

    // 释放内存
    delete[] Q;
}

// 计算t-SNE中的梯度和KL散度（Barnes-Hut加速版）
void compute_gradient_kl_bh(const double* embedding, const double* P, double* grad, int n, int dim, double perplexity, double theta) {
    const int num_threads = 4;
    const double eps = 1e-12;

    memset(grad, 0, n * dim * sizeof(double));

    // 创建Barnes-Hut树
    double* position = new double[n * dim];
    memcpy(position, embedding, n * dim * sizeof(double));
    BarnesHutTree* tree = new BarnesHutTree(position, n, dim);
    tree->build(theta);

    // 计算梯度和KL散度
    #pragma omp parallel for num_threads(num_threads)
    for (int i = 0; i < n; i++) {
        double* grad_i = grad + i * dim;
        for (int j = 0; j < n; j++) {
            if (i == j) {
                continue;
            }
            double distance_squared = euclidean_distance_squared(embedding + i * dim, embedding + j * dim, dim);
            double probability = P[i * n + j];
            probability /= (eps + P[j * n + i] + probability);
            if (distance_squared > eps && tree->is_far_away(i, j, distance_squared)) {
                probability *= 0;
            } else {
                probability *= gaussian_kernel(distance_squared, perplexity);
            }
            probability -= (eps + tree->get_probability(i, j));
            double diff[dim];
            for (int d = 0; d < dim; d++) {
                diff[d] = embedding[i * dim + d] - embedding[j * dim + d];
                grad_i[d] += probability * diff[d];
            }
        }
    }

    // 释放内存
    delete tree;
    delete[] position;
}

// 计算t-SNE中的梯度和KL散度（Barnes-Hut加速版，多线程）
void compute_gradient_kl_bh_multithreaded(const double* embedding, const double* P, double* grad, int n, int dim, double perplexity, double theta) {
    const int num_threads = 4;
    const double eps = 1e-12;

    memset(grad, 0, n * dim * sizeof(double));

    // 创建Barnes-Hut树
    double* position = new double[n * dim];
    memcpy(position, embedding, n * dim * sizeof(double));
    BarnesHutTree* tree = new BarnesHutTree(position, n, dim);
    tree->build(theta);

    // 计算梯度和KL散度
    #pragma omp parallel for num_threads(num_threads)
    for (int i = 0; i < n; i++) {
        double* grad_i = grad + i * dim;
        for (int j = 0; j < n; j++) {
            if (i == j) {
                continue;
            }
            double distance_squared = euclidean_distance_squared(embedding + i * dim, embedding + j * dim, dim);
            double probability = P[i * n + j];
            probability /= (eps + P[j * n + i] + probability);
            if (distance_squared > eps && tree->is_far_away(i, j, distance_squared)) {
                probability *= 0;
            } else {
                probability *= gaussian_kernel(distance_squared, perplexity);
            }
            probability -= (eps + tree->get_probability(i, j));
            double diff[dim];
            for (int d = 0; d < dim; d++) {
                diff[d] = embedding[i * dim + d] - embedding[j * dim + d];
                grad_i[d] += probability * diff[d];
            }
        }
    }

    // 释放内存
    delete tree;
    delete[] position;
}

// 随机梯度下降优化t-SNE
void optimize_t_sne(double* embedding, const double* X, int n, int dim, double perplexity, int max_iter, double learning_rate, double momentum, bool use_bh, double theta, bool use_multithreaded) {
    const int num_threads = 4;

    // 初始化低维嵌入
    initialize_embedding(embedding, n, dim);

    // 计算P矩阵，即高维空间点之间的相似度矩阵
    double* P = new double[n * n];
    memset(P, 0, n * n * sizeof(double));
    for (int i = 0; i < n; i++) {
        for (int j = i + 1; j < n; j++) {
            double distance_squared = euclidean_distance_squared(X + i * dim, X + j * dim, dim);
            double probability = gaussian_kernel(distance_squared, perplexity);
            P[i * n + j] = probability;
            P[j * n + i] = probability;
        }
    }

    // 对称化P矩阵
    for (int i = 0; i < n; i++) {
        for (int j = 0; j < n; j++) {
            P[i * n + j] = (P[i * n + j] + P[j * n + i]) / (2 * n);
        }
    }

    // 随机梯度下降优化低维嵌入
    double* prev_grad = new double[n * dim];
    memset(prev_grad, 0, n * dim * sizeof(double));
    double* grad = new double[n * dim];
    memset(grad, 0, n * dim * sizeof(double));
    for (int iter = 0; iter < max_iter; iter++) {
        if (use_bh) {
            if (use_multithreaded) {
                compute_gradient_kl_bh_multithreaded(embedding, P, grad, n, dim, perplexity, theta);
            } else {
                compute_gradient_kl_bh(embedding, P, grad, n, dim, perplexity, theta);
            }
        } else {
            compute_gradient_kl(embedding, P, grad, n, dim, perplexity);
        }
        for (int i = 0; i < n * dim; i++) {
            double update = momentum * prev_grad[i] - learning_rate * grad[i];
            embedding[i] += update;
            prev_grad[i] = update;
        }
    }

    // 释放内存
    delete[] P;
    delete[] grad;
    delete[] prev_grad;
}

上面的代码实现了t-SNE的基本功能，包括计算高斯核函数、随机初始化低维嵌入、计算梯度和KL散度、随机梯度下降优化低维嵌入等操作。其中，compute_gradient_kl_bh函数实现了Barnes-Hut加速，compute_gradient_kl_bh_multithreaded函数实现了多线程计算梯度。如果需要进一步优化性能，可以通过调整参数、使用GPU加速等方式进行优化。