# Difference between revisions of "measuring statistical dependence with Hilbert-Schmidt norms"

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== Cross-Covariance Operators == | == 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 | '''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 | ||

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+ | <math>||\mathit{C}||^2_{HS}:=\sum_{i,j}<\mathit{C}v_i,u_j>_\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. | |

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− | + | '''Hilbert-Schmidt Operator''' is defined based on the definition of Hilbert Schmidt norm as | |

− | + | <math><\mathit{C},\mathit{D}>{HS}:=\sum_{i,j}<\mathit{C}v_i,u_j>_\mathcal{F}<\mathit{D}v_i,u_j>_\mathcal{F}</math> | |

− | <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 |

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+ | <math>(f\otimes g)h:=f<g,h>_\mathcal{G}</math> for all <math>h\in \mathcal{G}</math> | ||

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

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+ | <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> | ||

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## Revision as of 17:10, 14 August 2013

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](\mathscr{X}\times\mathscr{Y},\mathscr{\Gamma}\times\mathscr{\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]

## 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.