度量两个矩阵的相似性

问题

如果有2个矩阵$Y\in R^{n\times q}$ 和 $Z\in R^{n\times c}$ ,如何度量两个矩阵的相似性？

回答

采用The Hilbert-Schmidt Independence Criterion (HSIC)进行度量。
$$HSIC(Z,Y)=(n-1{)}^{-2}Tr(HKHQ)$$
其中，$K=Z{Z}^T$ and $Q=Y{Y}^T$ are the inner
product of instances in Z and Y respectively and $H=I-\frac{1}{n}e{e}^T$ and
$I$ denote the identity matrix(单位矩阵),and $e$ be a vector of all ones.

理解

• 计算结果值越大，表示越接近。

参考：
[1] Huang R, Wu Z. Multi-label feature selection via manifold regularization and dependence maximization[J]. Pattern Recognition, 2021, 120: 108149.
[2] Gretton A , Bousquet O , Smola A , et al. Measuring Statistical Dependence with Hilbert-Schmidt Norms[J]. Springer, Berlin, Heidelberg, 2005.