On Brownian Distance Covariance and High Dimensional Data

By Michael R. Kosorok. 2009

964 citations recorded by the source corpus.

We discuss briefly the very interesting concept of Brownian distance covariance developed by Székely and Rizzo (2009) and describe two possible extensions. The first extension is for high dimensional data that can be coerced into a Hilbert space, including certain high throughput screening and functional data settings. The second extension involves very simple modifications that may yield increased power in some settings. We commend Székely and Rizzo for their very interesting work and recognize that this general idea has potential to have a large impact on the way in which statisticians evaluate dependency in data.

Subfields: Finance, Statistics and Probability, Statistical Learning Theory

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Citations in the corpus

Kernel Mean Embedding of Distributions: A Review and Beyonds (2016, background): “The distance covariance was later introduced in Székely et al. (2007), Székely and Rizzo (2009) for an independence test as a weighted L2-distance between characteristic functions of the joint and product distributions.”
NoPeek: Information leakage reduction to share activations in distributed deep learning (2020, methodology): “Our approach is based on minimizing a statistical dependency measure called distance correlation [10, 11, 12, 13, 14] between raw data and any intermediary communications across the clients or server participating in distributed deep learning.”
Supervised Dimensionality Reduction via Distance Correlation Maximization (2016, background): “Distance Correlation introduced by Szekely et al. [2007] and Székely et al. [2009], Székely and Rizzo [2012, 2013] is a measure nonlinear dependencies between random vectors of arbitrary dimensions.”
Kernel‐based tests for joint independence (2016, methodology): “received considerable attention using kernel-based methods (Gretton et al., 2005, 2007), and other related approaches for estimating or testing pairwise (in-)dependence include distance correlations (Székely and Rizzo, 2009, 2014), rank-based correlations (Bergsma and Dassios, 2014; Leung and Drton,”
On the Decreasing Power of Kernel and Distance Based Nonparametric Hypothesis Tests in High Dimensions (2014, methodology): “A related set of approaches were developed in parallel based on pairwise distances between points, as exemplified for independence testing by distance correlation, introduced in (Székely, Rizzo, and Bakirov 2007) and further discussed or extended in (Székely and Rizzo 2009; Lyons 2013; Székely and R”
A Kernel Test for Three-Variable Interactions (2013, methodology): “When a bounded characteristic kernel is used, the above tests are consistent against all alternatives, and their alternative interpretation is as a generalization [29, 3] of energy distance [30, 31] and distance covariance [2, 32].”
Equivalence of distance-based and RKHS-based statistics in hypothesis testing (2012, background): “In the discussion of [Sz´ekely and Rizzo (2009), Gretton, Fukumizu and Sriperumbudur (2009), page 1289] ﬁrst explored this link in the context of independence testing, and found that interpreting the distance-based independence statistic as a kernel statistic is not straightforward, since Bochner’s…”
Hypothesis testing using pairwise distances and associated kernels (2012, methodology): “A related dependence measure between vectors of high dimension is the distance covariance (Székely et al., 2007; Székely & Rizzo, 2009), and the resulting test is again consistent for variables with bounded first moment. The distance covariance has had a major impact in the statistics community, wit”
Cross-Modality Distillation: A Case for Conditional Generative Adversarial Networks (2018, positive): “Cross-Modal Distillation (Distance Correlation [20] + Classifier Loss) 87.”
The exact equivalence of distance and kernel methods in hypothesis testing (2018, neutral): “̂Hsic(X, Y) = Dcov(X, Y) = E[k(X,X�)k(Y , Y �)] + E[k(X,X�)]E[k(Y , Y �)] − 2E[k(X,X�)k(Y , Y ��)], 402 C. Shen, J. T. Vogelstein 1 3”
Multivariate Rank-Based Distribution-Free Nonparametric Testing Using Measure Transportation (2019, methodology): “significantly due to the seminal work by Szekely and co-authors (see [7, 142, 140]), where the notion of distance covariance was introduced; recall that in Section 3.”
Detecting Novel Associations in Large Datasets (2011, methodology): “Other methods such as mutual information estimators (12-14), maximal correlation (8, 15), principal curve-based methods (16-19)(20), distance correlation (21), and the Spearman rank correlation coefficient all detect broader classes of relationships.”
Sufficient Dimension Reduction via Distance Covariance (2016, methodology): “We use a novel geometric idea in the proof of Lemma A.1 to overcome the difficulty, and then combine proposition 4.3 in Cook (1998) and the third property in theorem 3 in Székely and Rizzo (2009) to reach the conclusions.”
Large-scale kernel methods for independence testing (2016, methodology): “2 A straightforward estimator of dCor (Székely et al. 2007; Székely and Rizzo 2009) is then given by normalising Ξb(z) by the Frobenius norms of H Kx H and H Ky H , i.”