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The exact equivalence of distance and kernel methods in hypothesis testing

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  • Cencheng Shen

    (University of Delaware)

  • Joshua T. Vogelstein

    (Johns Hopkins University
    Johns Hopkins University)

Abstract

Distance correlation and Hilbert-Schmidt independence criterion are widely used for independence testing, two-sample testing, and many inference tasks in statistics and machine learning. These two methods are tightly related, yet are treated as two different entities in the majority of existing literature. In this paper, we propose a simple and elegant bijection between metric and kernel. The bijective transformation better preserves the similarity structure, allows distance correlation and Hilbert-Schmidt independence criterion to be always the same for hypothesis testing, streamlines the code base for implementation, and enables a rich literature of distance-based and kernel-based methodologies to directly communicate with each other.

Suggested Citation

  • Cencheng Shen & Joshua T. Vogelstein, 2021. "The exact equivalence of distance and kernel methods in hypothesis testing," AStA Advances in Statistical Analysis, Springer;German Statistical Society, vol. 105(3), pages 385-403, September.
    • Handle: RePEc:spr:alstar:v:105:y:2021:i:3:d:10.1007_s10182-020-00378-1
    DOI: 10.1007/s10182-020-00378-1
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    References listed on IDEAS

    as
    1. Cencheng Shen & Carey E. Priebe & Joshua T. Vogelstein, 2020. "From Distance Correlation to Multiscale Graph Correlation," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 115(529), pages 280-291, January.
    2. Zhou Zhou, 2012. "Measuring nonlinear dependence in time‐series, a distance correlation approach," Journal of Time Series Analysis, Wiley Blackwell, vol. 33(3), pages 438-457, May.
    3. Runze Li & Wei Zhong & Liping Zhu, 2012. "Feature Screening via Distance Correlation Learning," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 107(499), pages 1129-1139, September.
    4. K Fokianos & M Pitsillou, 2018. "Testing independence for multivariate time series via the auto-distance correlation matrix," Biometrika, Biometrika Trust, vol. 105(2), pages 337-352.
    5. Xueqin Wang & Wenliang Pan & Wenhao Hu & Yuan Tian & Heping Zhang, 2015. "Conditional Distance Correlation," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 110(512), pages 1726-1734, December.
    6. Youjin Lee & Cencheng Shen & Carey E Priebe & Joshua T Vogelstein, 2019. "Network dependence testing via diffusion maps and distance-based correlations," Biometrika, Biometrika Trust, vol. 106(4), pages 857-873.
    7. Liping Zhu & Kai Xu & Runze Li & Wei Zhong, 2017. "Projection correlation between two random vectors," Biometrika, Biometrika Trust, vol. 104(4), pages 829-843.
    8. Ruth Heller & Yair Heller & Malka Gorfine, 2013. "A consistent multivariate test of association based on ranks of distances," Biometrika, Biometrika Trust, vol. 100(2), pages 503-510.
    9. Gabor J. Szekely & Maria L. Rizzo, 2005. "Hierarchical Clustering via Joint Between-Within Distances: Extending Ward's Minimum Variance Method," Journal of Classification, Springer;The Classification Society, vol. 22(2), pages 151-183, September.
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    Cited by:

    1. Xu, Kai & Cheng, Qing, 2024. "Test of conditional independence in factor models via Hilbert–Schmidt independence criterion," Journal of Multivariate Analysis, Elsevier, vol. 199(C).

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