Kernel Methods for Measuring Independence

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

451 citations recorded by the source corpus.

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In Statistical Learning Theory

Citations in the corpus

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).”
Causal Generative Neural Networks (2017, methodology): “We compare PC-Gaussian, which employs a Gaussian conditional independence test on Fisher z-transformations, and PC-HSIC, which uses the HSIC conditional independence test with the Gamma approximation [Gretton et al. , 2005].”
Nonparametric Independence Testing for Small Sample Sizes (2014, methodology): “The most important property is: when the kernels k, l are “characteristic”, then the corresponding population statistic ‖ΣXY ‖2HS is zero iff X,Y are independent [Gretton et al., 2005a].”
Kernel Mean Estimation and Stein's Effect (2013, positive): “As a result, there have been successful applications that benefit from the kernel mean representation, namely, a maximum mean discrepancy (MMD) [8], kernel dependency measure [9], kernel two-sample-test [10], Hilbert space embedding of HMMs [11], and kernel Bayes rule [12].”
Hypothesis testing using pairwise distances and associated kernels (2012, methodology): “In the case of independence testing, we are given i.i.d. samples z = {(xi, yi)} m i=1 ∼ PXY , and the resulting V-statistic estimate (HSIC) is (Gretton et al., 2005a; 2008) HSIC(z; kX , kY) = 1 m2 Tr(KXHKYH), (12) where KX , KY and H are m × m matrices given by (KX )ij := kX (xi, xj), (KY)ij :=…”
Hilbert Space Embeddings and Metrics on Probability Measures (2009, background): “, 2008); as well as in machine learning applications including kernel independent component analysis (Bach and Jordan, 2002; Gretton et al., 2005) and kernel based dimensionality reduction for supervised learning (Fukumizu et al.”
A Kernel Method for the Two-Sample-Problem (2006, methodology): “Harchaoui et al. (2008) build on the work of Gretton et al.”
Kernel Distribution Embeddings: Universal Kernels, Characteristic Kernels and Kernel Metrics on Distributions (2016, background): “…ranging from homogeneity (Gretton et al., 2007), distribution comparison (Gretton et al., 2007, 2012) and (conditional) independence tests (Gretton et al., 2005, 2008; Fukumizu et al., 2008; Gretton and Gy¨orﬁ, 2010; Lopez-Paz et al., 2013) to generative adversarial networks (Dziugaite et…”
An Updated Literature Review of Distance Correlation and Its Applications to Time Series (2017, unknown): “They further establish the equivalence of distance covariance measures on semimetric spaces of negative type (as in (13)) and the Hilbert-Schmidt independence criterion (HSIC), see Gretton et al. (2005, 2008).”
A Survey of Multi-View Representation Learning (2016, unknown): “As a non-linear extension of CCA, kernel CCA has been successfully applied in many situations, including independent component analysis [2], cross-media information retrieval [3, 84, 85], computational biology [3, 86, 87], multi-view clustering [46, 88], acoustic feature learning [89, 90], and stati”
Information-geometric approach to inferring causal directions (2012, methodology): “All methods, except for GPI, employ the HSIC independence test [25] for accepting or rejecting the fitted models and use the maximum of the two HSIC p-values (where each p-value corresponds to a possible causal direction) as confidence estimate.”
On Causal Discovery with Cyclic Additive Noise Models (2011, methodology): “In each case, we calculated the p -value for independence of the two noise variables using the HSIC (Hilbert-Schmidt Independence Criterion) test [15]; for p -values substantially above 0 (say larger than 1% ), we do not reject the null hypothesis of independence and hence accept the model as…”
Multivariate Rank-Based Distribution-Free Nonparametric Testing Using Measure Transportation (2019, unknown): “[52] Gretton, A.”