Factorized Graph Matching

最新推荐文章于 2024-12-13 11:46:36 发布

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本文通过详细的数学推导展示了如何利用矩阵运算属性简化特定形式的矩阵表达式，特别是涉及Kronecker乘积、迹（trace）及向量化操作的计算。

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{
推导：7式
(3) $vec(X)^TKvec(X)$
(6) $K = diag(vec(K_p))+(G_2\otimes G_1)diag(vec(K_q))(H_2\otimes H_1)^T$
将(6)式代入(3)式得到:

v e c (X) T (d i a g (v e c (K p)) + (G 2 \otimes G 1) d i a g (v e c (K q)) (H 2 \otimes H 1) T) v e c (X) = v e c (X) T d i a g (v e c (K p)) v e c (X) + v e c (X) T (G 2 \otimes G 1) d i a g (v e c (K q)) (H 2 \otimes H 1) T v e c (X) = v e c (K p) T v e c (X \circ X) + v e c (G T 1 X G 2) T d i a g (v e c (K q)) v e c (H T 1 X H 2) ① = t r (K T p X) + v e c (K q) T v e c (G T 1 X G 2 \circ H T 1 X H 2) ② = t r (K T p X) + t r (K T q (G T 1 X G 2 \circ H T 1 X H 2))

$\begin{align} &vec(X)^T(diag(vec(K_p))+(G_2\otimes G_1)diag(vec(K_q))(H_2\otimes H_1)^T)vec(X)\\ &=vec(X)^Tdiag(vec(K_p))vec(X)+vec(X)^T(G_2\otimes G_1)diag(vec(K_q))(H_2\otimes H_1)^Tvec(X)\\ &=vec(K_p)^Tvec(X\circ X)+vec(G_1^TXG_2)^Tdiag(vec(K_q))vec(H_1^TXH_2) ①\\ &=tr(K_p^TX)+vec(K_q)^Tvec(G_1^TXG_2\circ H_1^TXH_2)②\\ &=tr(K_p^TX)+tr(K_q^T(G_1^TXG_2\circ H_1^TXH_2)) \end{align}$

①
$(A\otimes B)^T=A^T \otimes B^T$ (引用于matrix cookbook (510))
$vec(AXB)=(B^T\otimes A)vec(X)$ (引用于matrix cookbook (520))
②
$Tr(A^TB)=vec(A)^Tvec(B)$ (引用于matrix cookbook (521))
}