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_j\gt _\mathcal{F}^2 }[/math]

Where [math]\displaystyle{ v_i,u_j }[/math] are orthonormal bases of [math]\displaystyle{ \mathcal{G} }[/math] and [math]\displaystyle{ \mathcal{F} }[/math] respectively.

Hilbert-Schmidt Operator is defined based on the definition of Hilbert Schmidt norm as

[math]\displaystyle{ \lt \mathit{C},\mathit{D}\gt {HS}:=\sum_{i,j}\lt \mathit{C}v_i,u_j\gt _\mathcal{F}\lt \mathit{D}v_i,u_j\gt _\mathcal{F} }[/math]

Tensor Product. Let [math]\displaystyle{ f\in \mathcal{F} }[/math] and [math]\displaystyle{ g\in \mathcal{G} }[/math]. The tensor product operator [math]\displaystyle{ f\otimes g:\mathcal{G}\to \mathcal{F} }[/math] is defined as

[math]\displaystyle{ (f\otimes g)h:=f\lt g,h\gt _\mathcal{G} }[/math] for all [math]\displaystyle{ h\in \mathcal{G} }[/math]

Cross-Covariance Operator associated with the joint measure [math]\displaystyle{ p_{x,y} }[/math] on [math]\displaystyle{ (\mathcal{X}\times\mathcal{Y},\Gamma\times\Lambda) }[/math] is a linear operator [math]\displaystyle{ C_{xy}:\mathcal{G}\to \mathcal{F} }[/math] defined as

[math]\displaystyle{ C_{xy}:=E_{x,y}[(\theta (x)-\mu_x)\otimes (\psi (y)-\mu_y)]=E_{x,y}[\theta (x)\otimes \psi (y)]-\mu_x\otimes\mu_y }[/math]

Hilbert-Schmidt Independence Criterion

Given separable RKHSs [math]\displaystyle{ \mathcal{F},\mathcal{G} }[/math] and a joint measure [math]\displaystyle{ p_{xy} }[/math] over [math]\displaystyle{ (\mathcal{X}\times\mathcal{Y},\Gamma\times\Lambda) }[/math], HSIC is defined as the squared HS-norm of the associated cross-covariance operator [math]\displaystyle{ C_{xy} }[/math]:

[math]\displaystyle{ HSIC(p_{xy},\mathcal{F},\mathcal{G}):=||C_{xy}||_{HS}^2 }[/math]

According to Gretton et al., the largest singular value [math]\displaystyle{ ||C_{xy}||_{HS}=0 }[/math] if and only if x and y are independent. Since [math]\displaystyle{ ||C_{xy}=0||_S }[/math] if and only if [math]\displaystyle{ ||C_{xy}||_{HS}=0 }[/math], so [math]\displaystyle{ ||C_{xy}||_{HS}=0 }[/math] if and only if x and y are independent. Therefore, HSIC can be used as a independence criteria.

Estimator of HSIC

Let [math]\displaystyle{ Z:={(x_1,y_1),\dots,(x_m,y_m)}\subseteq \mathcal{X}\times\mathcal{Y} }[/math] be a series of m independent observations drawn from [math]\displaystyle{ p_{xy} }[/math]. An empirical estimator of HSIC, written HSIC(Z,\mathcal{F},\mathcal{G}) is given by

[math]\displaystyle{ HSIC(Z,\mathcal{F},\mathcal{G}):=(m-1)^{-2}trKHLH }[/math]

where [math]\displaystyle{ H,K,L\in \mathbb{R}^{m\times m},K_{ij}:=k(x_i,x_j),L_{i,j}:=l(y_i,y_j) and H_{ij}:=\delta_{ij}-m^{-1} }[/math]. It can be proved that the bias of the empirical HSIC is at [math]\displaystyle{ \mathit{O}(m^{-1}) }[/math].

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.