计算对称归一化拉普拉斯矩阵的问题

Question

我在python中计算对称归一化拉普拉斯矩阵时发现了一些问题。 假设有矩阵S及其对角度矩阵D：

    [ [ 1 , 0.5, 0.2]        [ [1.7, 0, 0 ]
S =   [0.5,  1 , 0.5]     D =  [ 0 , 2, 0 ] 
      [0.2, 0.5,  1 ] ]        [ 0 , 0,1.7] ]

当计算 L 时

我得到这个结果：

       [[ 0.41176471 -0.27116307 -0.11764706]
    L = [-0.27116307  0.5        -0.27116307]
        [-0.11764706 -0.27116307  0.41176471]]

使用此代码：

S = np.array([[1,0.5,0.2],[0.5,1,0.5],[0.2,0.5,1]])

print("Similiarity Matrix: \n",S)
print("\n\n")


D = np.zeros((len(S), len(S)))
#H = np.sum(G[0])
for id, x in enumerate(S):
    D[id][id] = np.sum(x)

I = np.identity(len(S))

L = I - ((sqrtm(inv(D))).dot(S)).dot(sqrtm(inv(D)))
print("\n\n")
print("Laplacian normalized: \n",L)

这与使用返回的函数csgraph.laplacian(S, normed=True) 不同：

       [[[ 1.        -0.5976143  -0.28571429]
    L = [-0.5976143   1.         -0.5976143 ]
        [-0.28571429 -0.5976143   1.        ]]

为什么会这样？我是不是做错了什么？

Answer 1

我注意到csgraph.laplacian 返回的未归一化矩阵和归一化矩阵之间的比率与未归一化矩阵和您的L 的比率密切相关：

In [20]: csgraph.laplacian(S, normed=False) / L - 1
Out[20]:
array([[0.7       , 0.84390889, 0.7       ],
       [0.84390889, 1.        , 0.84390889],
       [0.7       , 0.84390889, 0.7       ]])

In [21]: csgraph.laplacian(S, normed=False) / csgraph.laplacian(S, normed=True)
Out[21]:
array([[0.7       , 0.83666003, 0.7       ],
       [0.83666003, 1.        , 0.83666003],
       [0.7       , 0.83666003, 0.7       ]])

0.84390889 ≠ 0.83666003 但其他数字匹配。差异可能仅仅是归一化造成的吗？

Answer 2

那是因为你在 S 的对角线上有 1：

# weighted adjacency
S = np.array([[1,0.5,0.2],[0.5,1,0.5],[0.2,0.5,1]])

np.fill_diagonal(S, 0.0)

# strength diagonal matrix
D = np.diag(np.sum(S,axis=1))

# identity
I = np.identity(S.shape[0])

# D^{-1/2} matrix
D_inv_sqrt = np.linalg.inv(np.sqrt(D))

L = I - np.dot(D_inv_sqrt, S).dot(D_inv_sqrt)

L

array([[ 1.        , -0.5976143 , -0.28571429],
       [-0.5976143 ,  1.        , -0.5976143 ],
       [-0.28571429, -0.5976143 ,  1.        ]])