# measuring statistical dependence with Hilbert-Schmidt norms

This is another very popular kernel-based approach fro detecting dependence which is called HSIC(Hilbert-Schmidt Independence Criteria). It's based on the eigenspectrum of covariance operators in reproducing kernel Hilbert spaces(RKHSs).This approach is simple and no user-defined regularisation is needed. Exponential convergence is guaranteed, so convergence is fast.

## Background

Before the proposal of HSIC, there are already a few kernel-based independence detecting methods. Bach[] proposed a regularised correlation operator which is derived from the covariance and cross-covariance operators, and its largest singular value was used as a static to test independence. Gretton et al.[] used the largest singular value of the cross-covariance operator which resulted constrained covariance(COCO). HSIC is a extension of the concept COCO by using the entire spectrum of cross-covariance operator to determine when all its singular values are zero rather than just looking the largest singular value.

## Cross-Covariance Operators

Cross-covariance operator is first propose by (Baker,1973). It can be used to measure the relations between probability measures on two RKHSs. Define two RKHSs $H_1$ and $H_2$ with inner product $\lt .,.\gt _1$, $\lt .,.\gt _2$. A probability measure $\mu_i$ on $H_i,i=1,2$ that satisfies

$\int_{H_i}||x||_i^2d\mu_i(x)\lt \infty$

defines an operator $R_i$ in $H_i$ by

$\lt R_iu,v\gt =\int_{H_i}\lt x-m_i,u\gt _i\lt x-m_i,v\gt _id\mu_i(x)$

$R_i$ is called covariance operator, if u and v are in different RKHS, then $R_i$ is called cross-covariance operator.

## References

[1] Gretton, Arthur, et al. "Measuring statistical dependence with Hilbert-Schmidt norms." Algorithmic learning theory. Springer Berlin Heidelberg, 2005.

[2] Fukumizu, Kenji, Francis R. Bach, and Michael I. Jordan. "Kernel dimension reduction in regression." The Annals of Statistics 37.4 (2009): 1871-1905.

[3] Bach, Francis R., and Michael I. Jordan. "Kernel independent component analysis." The Journal of Machine Learning Research 3 (2003): 1-48.

[4] Baker, Charles R. "Joint measures and cross-covariance operators." Transactions of the American Mathematical Society 186 (1973): 273-289.