Maximum Mean Discrepancy, Hilbert-Schmidt Independence Criterion

August 21, 2016 — February 29, 2024

approximation
functional analysis
Hilbert space
measure
metrics
nonparametric
optimization
probability
statistics

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An integral probability metric. The intersection of reproducing kernel methods, dependence tests, and probability metrics; where we use a kernel embedding to cleverly measure differences between probability distributions, typically an RKHS embedding, but any old Hilbert space will do.

Can be estimated from samples only, which is neat.

A mere placeholder. For a thorough treatment, see the canonical references (Gretton et al. 2008; Gretton, Borgwardt, et al. 2012).

Figure 1

1 Tutorial

Arthur Gretton, Dougal Sutherland, Wittawat Jitkrittum presentation: Interpretable Comparison of Distributions and Models.

Danica Sutherland’s explanation is clear.

Pierre Alquier’s post Universal estimation with Maximum Mean Discrepancy (MMD) shows how to use MMD in a robust nonparametric estimator.

Gaël Varoquaux’ introduction is friendly and illustrated, Comparing distributions: Kernels estimate good representations, l1 distances give good tests based on (Scetbon and Varoquaux 2019).

2 Hilbert-Schmidt Independence Criterion

The HSIC is the application of the MMD to dependence testing, AFAICT.

3 Connection to optimal transport losses

Husain (2020)’s results connect IPMs to transport metrics and regularisation theory, and classification.

Feydy et al. (2019) connects MMD to optimal transport losses.

Arbel et al. (2019) also looks pertinent and has some connections to Wasserstein gradient flow, which is a thing. That looks super interesting and is continued over a MMD flows

4 Connection to kernelized Stein discrepancy

TBD. See Stein VGD.

5 Choice of kernel

Hmm. See Gretton, Sriperumbudur, et al. (2012).

6 Tooling

MMD is included in the ITE toolbox (estimators).

6.1 GeomLoss

The GeomLoss library provides efficient GPU implementations for:

It is hosted on GitHub and distributed under the permissive MIT license. pypi pepy

GeomLoss functions are available through the custom PyTorch layers SamplesLoss, ImagesLoss and VolumesLoss which allow you to work with weighted point clouds (of any dimension), density maps and volumetric segmentation masks.

7 Incoming

8 References

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Alquier, and Gerber. 2024. “Universal Robust Regression via Maximum Mean Discrepancy.” Biometrika.
Arbel, Korba, Salim, et al. 2019. “Maximum Mean Discrepancy Gradient Flow.” In Proceedings of the 33rd International Conference on Neural Information Processing Systems.
Arras, Azmoodeh, Poly, et al. 2017. “A Bound on the 2-Wasserstein Distance Between Linear Combinations of Independent Random Variables.” arXiv:1704.01376 [Math].
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Briol, Barp, Duncan, et al. 2019. “Statistical Inference for Generative Models with Maximum Mean Discrepancy.”
Cherief-Abdellatif, and Alquier. 2020. “MMD-Bayes: Robust Bayesian Estimation via Maximum Mean Discrepancy.” In Proceedings of The 2nd Symposium on Advances in Approximate Bayesian Inference.
Deb, Ghosal, and Sen. 2020. “Measuring Association on Topological Spaces Using Kernels and Geometric Graphs.” arXiv:2010.01768 [Math, Stat].
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Gretton, Borgwardt, Rasch, et al. 2012. “A Kernel Two-Sample Test.” The Journal of Machine Learning Research.
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