拉普拉斯特征映射代码

以下是拉普拉斯特征映射的 Python 代码实现：

import numpy as np
from sklearn.preprocessing import normalize
from sklearn.neighbors import NearestNeighbors
from scipy.sparse.linalg import eigsh

def laplacian_eigenmaps(X, n_components, k=5, gamma=0.1):
    """
    Laplacian Eigenmaps algorithm for dimensionality reduction.
    
    Parameters:
    -----------
    X: numpy array or sparse matrix, shape (n_samples, n_features)
        The input data to reduce dimensionality.
    n_components: int
        The number of components to keep.
    k: int, optional (default=5)
        Number of neighbors to use when constructing the adjacency matrix.
    gamma: float, optional (default=0.1)
        Parameter controlling the strength of the affinity matrix.
        
    Returns:
    --------
    Y: numpy array, shape (n_samples, n_components)
        The reduced data.
    """
    
    # Compute pairwise euclidean distances
    nn = NearestNeighbors(n_neighbors=k+1)
    nn.fit(X)
    distances, indices = nn.kneighbors(X)
    W = np.exp(-gamma * distances**2)
    np.fill_diagonal(W, 0)
    
    # Compute the degree matrix
    D = np.diag(np.sum(W, axis=1))
    
    # Compute the unnormalized Laplacian
    L = D - W
    
    # Compute the normalized Laplacian
    D_inv_sqrt = np.diag(1 / np.sqrt(np.sum(W, axis=1)))
    L_norm = D_inv_sqrt @ L @ D_inv_sqrt
    
    # Compute the eigenvectors and eigenvalues of L_norm
    eigenvalues, eigenvectors = eigsh(L_norm, k=n_components+1, which='SM')
    
    # Sort the eigenvectors by eigenvalues
    idx = eigenvalues.argsort()
    eigenvectors = eigenvectors[:, idx]
    
    # Remove the first eigenvector (eigenvalue=0)
    eigenvectors = eigenvectors[:, 1:n_components+1]
    
    # Normalize the eigenvectors
    Y = normalize(eigenvectors, norm='l2', axis=1)
    
    return Y

这个实现使用了 Scikit-Learn 的 `NearestNeighbors` 类来计算邻居之间的距离，使用了 SciPy 的 `eigsh` 函数来计算特征向量和特征值，并使用 Scikit-Learn 的 `normalize` 函数对特征向量进行归一化。