Python实现谱聚类算法

谱聚类是一种基于图论的聚类方法，适用于非凸、非凹、不规则形状的数据集。其基本思想是将数据集看作图的顶点集合，根据顶点之间的相似性构建边权重矩阵，进而求解谱分解，得到特征向量。通过对特征向量进行聚类，即可得到数据集的聚类结果。

Python实现谱聚类算法的具体步骤如下：

1. 建立数据集的相似性矩阵，通常使用高斯核函数计算相似度：

def similarity_matrix(X, sigma=1):
    n_samples = X.shape[0]
    W = np.zeros((n_samples, n_samples))
    for i in range(n_samples):
        for j in range(i+1, n_samples):
            d = np.linalg.norm(X[i] - X[j])
            W[i, j] = np.exp(-d**2 / (2*sigma**2))
            W[j, i] = W[i, j]
    return W

2. 计算拉普拉斯矩阵，有两种方式：

（1）标准拉普拉斯矩阵：$L = D - W$，其中$D$为度矩阵，$W$为相似性矩阵。

（2）对称归一化拉普拉斯矩阵：$L = I - D^{-1/2}WD^{-1/2}$。

def laplacian_matrix(W, type='unnormalized'):
    n_samples = W.shape[0]
    D = np.diag(np.sum(W, axis=1))
    if type == 'unnormalized':
        L = D - W
    elif type == 'symmetric':
        D_sqrt = np.sqrt(np.linalg.inv(D))
        L = np.dot(np.dot(D_sqrt, (D - W)), D_sqrt)
    return L

3. 对拉普拉斯矩阵进行谱分解，得到特征向量矩阵和特征值矩阵：

def spectral_decomposition(L, n_clusters):
    eigvals, eigvecs = np.linalg.eig(L)
    idx = eigvals.argsort()
    eigvecs = eigvecs[:, idx]
    eigvals = eigvals[idx]
    U = eigvecs[:, :n_clusters]
    return U

4. 对特征向量进行KMeans聚类：

from sklearn.cluster import KMeans

def spectral_clustering(X, n_clusters, sigma=1):
    W = similarity_matrix(X, sigma)
    L = laplacian_matrix(W, type='symmetric')
    U = spectral_decomposition(L, n_clusters)
    kmeans = KMeans(n_clusters=n_clusters)
    labels = kmeans.fit_predict(U)
    return labels

完整代码：

import numpy as np
from sklearn.cluster import KMeans

def similarity_matrix(X, sigma=1):
    n_samples = X.shape[0]
    W = np.zeros((n_samples, n_samples))
    for i in range(n_samples):
        for j in range(i+1, n_samples):
            d = np.linalg.norm(X[i] - X[j])
            W[i, j] = np.exp(-d**2 / (2*sigma**2))
            W[j, i] = W[i, j]
    return W

def laplacian_matrix(W, type='unnormalized'):
    n_samples = W.shape[0]
    D = np.diag(np.sum(W, axis=1))
    if type == 'unnormalized':
        L = D - W
    elif type == 'symmetric':
        D_sqrt = np.sqrt(np.linalg.inv(D))
        L = np.dot(np.dot(D_sqrt, (D - W)), D_sqrt)
    return L

def spectral_decomposition(L, n_clusters):
    eigvals, eigvecs = np.linalg.eig(L)
    idx = eigvals.argsort()
    eigvecs = eigvecs[:, idx]
    eigvals = eigvals[idx]
    U = eigvecs[:, :n_clusters]
    return U

def spectral_clustering(X, n_clusters, sigma=1):
    W = similarity_matrix(X, sigma)
    L = laplacian_matrix(W, type='symmetric')
    U = spectral_decomposition(L, n_clusters)
    kmeans = KMeans(n_clusters=n_clusters)
    labels = kmeans.fit_predict(U)
    return labels