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

Hilbert-Schmidt Norm. Denote by [math]\displaystyle{ \mathit{C}:\mathcal{G}\to\mathcal{F} }[/math] a linear operator. Provided the sum converges, the HS norm of [math]\displaystyle{ \mathit{C} }[/math] is defined as [math]\displaystyle{ ||\mathit{C}||^2_{HS}:=\sum_{i,j}\lt \mathit{C}v_i,u_i\gt _\mathcal{F}^2 }[/math]

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 [math]\displaystyle{ H_1 }[/math] and [math]\displaystyle{ H_2 }[/math] with inner product [math]\displaystyle{ \lt .,.\gt _1 }[/math], [math]\displaystyle{ \lt .,.\gt _2 }[/math]. A probability measure [math]\displaystyle{ \mu_i }[/math] on [math]\displaystyle{ H_i,i=1,2 }[/math] that satisfies

[math]\displaystyle{ \int_{H_i}||x||_i^2d\mu_i(x)\lt \infty }[/math]

defines an operator [math]\displaystyle{ R_i }[/math] in [math]\displaystyle{ H_i }[/math] by

[math]\displaystyle{ \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) }[/math]

[math]\displaystyle{ R_i }[/math] is called covariance operator, if u and v are in different RKHS, then [math]\displaystyle{ R_i }[/math] 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.