Measuring Statistical Dependence with Hilbert-Schmidt Norms

By Arthur Gretton, Olivier Bousquet, Alex Smola, Bernhard Schölkopf. 2005

2,028 citations recorded by the source corpus.

Subfields: Artificial Intelligence, Computer Vision and Pattern Recognition, Statistics and Probability, Statistical Learning Theory, Signal Processing

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Citations in the corpus

Improved Techniques for Training GANs (2016, neutral): “3.1, is similar in spirit to approaches that use maximum mean discrepancy [7, 8, 9] to train generator networks [10, 11].”
Toward Causal Representation Learning (2021, background): “Recent progress in (conditional) independence testing heavily relies on kernel function classes to represent probability distributions in reproducing kernel Hilbert spaces [43], [61], [74], [91], [92], [193], [280].”
Kernel Mean Embedding of Distributions: A Review and Beyonds (2016, methodology): “Similar to the idea of HSIC described in §3.6, Fukumizu et al. (2008) proposes a nonparametric conditional dependence measure based on the normalized conditional cross-covariance operator VYX |Z := VYX − VYZVZX (4.26) where VYX is a unique bounded operator such that CY X = C1/2Y Y VY XC 1/2 XX (Bake”
Towards Causal Representation Learning (2021, positive): “progress in (conditional) independence testing heavily relies on kernel function classes to represent probability distributions in reproducing kernel Hilbert spaces [90, 91, 73, 278, 60, 191, 42].”
Causality for Machine Learning (2019, methodology): “Recent progress in (conditional) independence testing heavily relies on kernel function classes to represent probability distributions in reproducing kernel Hilbert spaces (Gretton et al., 2005a,b; Fukumizu et al., 2008; Zhang et al., 2011; Pfister et al., 2018b; Chalupka et al., 2018).”
Kernel‐based tests for joint independence (2016, methodology): “…tests date back to Feuerverger (1993) and Romano (1986) and have more recently received considerable attention by using kernel-based methods (Gretton et al., 2005, 2007), and other related approaches for estimating or testing pairwise (in)dependence including distance correlations (Székely…”
Bayesian Learning of Kernel Embeddings (2016, methodology): “The empirical versions of MMD and HSIC are used as test statistics for the two-sample ( H 0 : P = Q vs.”
Computing functions of random variables via reproducing kernel Hilbert space representations (2015, positive): “In all cases, we employ a Gaussian RBF kernel (26) whose bandwidth parameter is chosen using the median heuristic, setting σ to the median of the pairwise distances of distinct data points (Gretton et al. 2005).”
Distinguishing Cause from Effect Using Observational Data: Methods and Benchmarks (2014, methodology): “A.1 Consistency of HSIC We recapitulate the definitions and some asymptotic properties of the Hilbert Schmidt Independence Criterion (HSIC), following mostly the notations and terminology in (Gretton et al., 2005).”
A Wild Bootstrap for Degenerate Kernel Tests (2014, methodology): “This quantity is called Hilbert-Schmidt Independence Criterion (HSIC) [16, 17].”
A low variance consistent test of relative dependency (2014, methodology): “M L ] 1 variables and the source is computed using the Hilbert-Schmidt Independence Criterion [4, 5].”
On the Decreasing Power of Kernel and Distance Based Nonparametric Hypothesis Tests in High Dimensions (2014, methodology): “Indeed, corresponding to MMD, there exists a test statistic using kernels called HSIC, see (Gretton et al. 2005), for the independence testing problem.”
A Kernel Independence Test for Random Processes (2014, methodology): “We use as our test statistic the Hilbert-Schmidt Independence Criterion (HSIC) (Gretton et al., 2005; 2008b), which can be represented as the distance between embeddings of the joint distribution and the product of the marginals in a reproducing kernel Hilbert space (RKHS) (Gretton et al.”
A Kernel Test for Three-Variable Interactions (2013, background): “Much of the work in this area focuses on measuring or testing pairwise interaction: for instance, the Hilbert-Schmidt Independence Criterion (HSIC) or Distance Covariance [1, 2, 3], kernel canonical correlation [4, 5, 6], and mutual information [7].”
Hypothesis testing using pairwise distances and associated kernels (2012, methodology): “Second, results from Gretton et al. (2009a); Zhang et al. (2011) may be applied to get consistent two-sample and independence tests for the energy distance, without using bootstrap, which perform much better than the upper bound proposed by Székely et al. (2007) as an alternative to the bootstrap.”
Generative and Latent Mean Map Kernels (2010, methodology): “Also, while the EMMK previously has been used to measure and optimize dependence [5, 30, 4], it has not yet been used for classification.”