Sobolev Independence Criterion



  • Youssef Mroueh
  • Tom Sercu
  • Mattia Rigotti
  • Inkit Padhi
  • Cicero Nogueira dos Santos

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We propose the Sobolev Independence Criterion (SIC), an interpretable dependency measure between a high dimensional random variable X and a response variable Y . SIC decomposes to the sum of feature importance scores and hence can be used for nonlinear feature selection. SIC can be seen as a gradient regularized Integral Probability Metric (IPM) between the joint distribution of the two random variables and the product of their marginals. We use sparsity inducing gradient penalties to promote input sparsity of the critic of the IPM. In the kernel version we show that SIC can be cast as a convex optimization problem by introducing auxiliary variables that play an important role in feature selection as they are normalized feature importance scores. We then present a neural version of SIC where the critic is parameterized as a homogeneous neural network, improving its representation power as well as its interpretability. We conduct experiments validating SIC for feature selection in synthetic and real-world experiments. We show that SIC enables reliable and interpretable discoveries, when used in conjunction with the holdout randomization test and knockoffs to control the False Discovery Rate. Code is available at

This work was published in NeurIPS 2019.

Please cite our work using the BibTeX below.

title = {Sobolev Independence Criterion},
author = {Mroueh, Youssef and Sercu, Tom and Rigotti, Mattia and Padhi, Inkit and Nogueira dos Santos, Cicero},
booktitle = {Advances in Neural Information Processing Systems 32},
editor = {H. Wallach and H. Larochelle and A. Beygelzimer and F. d\textquotesingle Alch\'{e}-Buc and E. Fox and R. Garnett},
pages = {9505--9515},
year = {2019},
publisher = {Curran Associates, Inc.},
url = {}
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