增强经典多维尺度变换（CMDS）对无穷大距离矩阵的处理能力

import numpy as np
from sklearn.metrics.pairwise import euclidean_distances

def cmds(X, n_dim, input_type='raw'):
    """
    Classical(linear) multidimensional scaling (MDS)

    Parameters
    ----------
    X: (d, n) array or (n,n) array
        input data. The data are placed in column-major order. 
        That is, samples are placed in the matrix (X) as column vectors
        d: dimension of points
        n: number of points

    n_dim: dimension of target space

    input_type: it indicates whether data are raw or distance
        - raw: raw data. (n,d) array. 
        - distance: precomputed distances between the data. (n,n) array.
    Returns
    -------
    Y: (n_dim, n) array. projected embeddings.
    evals: (n_dim) eigen values
    evecs: corresponding eigen vectors in column vectors
    """

    if input_type == 'distance':
        D = X
    elif input_type == 'raw':
        # Transpose X to (n, d) for euclidean_distances
        Xt = X.T
        D = euclidean_distances(Xt, Xt)
    else:
        raise ValueError("input_type must be 'raw' or 'distance'")

    # 检查距离矩阵中是否存在无穷大值，并进行替换
    if np.any(np.isinf(D)):
        # 将inf值替换为该数据类型所能表示的最大有限浮点数
        # 这样做可以避免在后续计算中因inf值导致错误，同时保留其“非常远”的语义
        D[np.isinf(D)] = np.finfo(D.dtype).max

    # Centering matrix
    n = D.shape[0]
    H = np.eye(n) - np.ones((n, n)) / n

    # Double-center the distance matrix
    # 注意：这里D**2是元素级的平方操作
    B = -0.5 * H @ (D**2) @ H

    # Eigen decomposition
    evals, evecs = np.linalg.eigh(B)

    # Sorting eigenvalues and eigenvectors in decreasing order
    sort_indices = np.argsort(evals)[::-1]
    evals = evals[sort_indices]
    evecs = evecs[:, sort_indices]

    # Selecting top n_dim eigenvectors
    evecs_selected = evecs[:, :n_dim]

    # Projecting data to the new space
    # 确保特征值非负，因为它们理论上应代表方差
    # 实际应用中，由于数值精度或非欧氏距离，可能出现微小负特征值，
    # 但对于CMDS，通常只考虑正特征值。这里假设前n_dim特征值是有效的。
    valid_evals = np.maximum(0, evals[:n_dim]) # 避免负数开方
    Y = np.sqrt(np.diag(valid_evals)) @ evecs_selected.T

    return Y, evals, evecs