不调用sklearn.manifold ，实现ISOMAP降维度。

ISOMAP是一种基于流形学习的非线性降维算法，可以将高维数据映射到低维空间中。下面是一个简单的实现ISOMAP降维的代码：

1. 定义计算欧几里得距离的函数

def euclidean_distance(x1, x2):
    return np.sqrt(np.sum((x1 - x2) ** 2))

2. 构建邻接矩阵

def construct_adjacency_matrix(X, k):
    n = X.shape[0]
    W = np.zeros((n, n))
    for i in range(n):
        dist = []
        for j in range(n):
            if i != j:
                dist.append((j, euclidean_distance(X[i], X[j])))
        dist.sort(key=lambda x: x[1])
        for j in range(1, k+1):
            idx, d = dist[j]
            W[i][idx] = d
            W[idx][i] = d
    return W

3. 计算最短距离矩阵

def shortest_path(W):
    n = W.shape[0]
    D = np.full((n, n), np.inf)
    for i in range(n):
        D[i][i] = 0
        for j in range(n):
            if W[i][j] != 0:
                D[i][j] = W[i][j]
    for k in range(n):
        for i in range(n):
            for j in range(n):
                if D[i][j] > D[i][k] + D[k][j]:
                    D[i][j] = D[i][k] + D[k][j]
    return D

4. 根据最短距离矩阵计算嵌入矩阵

def isomap(X, k, n_components):
    W = construct_adjacency_matrix(X, k)
    D = shortest_path(W)
    B = -0.5 * (D ** 2 - np.sum(D ** 2, axis=0) / n + np.sum(D ** 2) / (n ** 2))
    eigvals, eigvecs = np.linalg.eig(B)
    idx = np.argsort(eigvals)[::-1][:n_components]
    eigvals, eigvecs = eigvals[idx], eigvecs[:, idx]
    Y = eigvecs * np.sqrt(eigvals)
    return Y.real

其中，参数`X`是原始数据，`k`是近邻数，`n_components`是降维后的维度。函数返回降维后的结果。