measuring statistical dependence with Hilbert-Schmidt norms

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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]\mathit{C}:\mathcal{G}\to\mathcal{F}[/math] a linear operator. Provided the sum converges, the HS norm of [math]\mathit{C}[/math] is defined as

[math]||\mathit{C}||^2_{HS}:=\sum_{i,j}\lt \mathit{C}v_i,u_j\gt _\mathcal{F}^2[/math]

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

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

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

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

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

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

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

According to Gretton et al., the largest singular value [math]||C_{xy}||_{HS}=0[/math] if and only if x and y are independent. Since [math]||C_{xy}=0||_S[/math] if and only if [math]||C_{xy}||_{HS}=0[/math], so [math]||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]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]p_{xy}[/math]. An empirical estimator of HSIC, written HSIC(Z,\mathcal{F},\mathcal{G}) is given by

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

where [math]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]\mathnormal{O}(m^{-1})[/math].


Independence Tests

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.