Transport maps

Inference by measure transport, low-dimensional coupling…



Maps moving one probability measure into another. Useful in their own right. If we assign costs to various maps and then prefer some to others then we are in the realm of optimal transport metrics.

References

Bonnotte, Nicolas. 2012. From Knothe’s Rearrangement to Brenier’s Optimal Transport Map.” arXiv.
Carlier, Guillaume, Alfred Galichon, and Filippo Santambrogio. 2008. From Knothe’s Transport to Brenier’s Map and a Continuation Method for Optimal Transport.” arXiv.
Cui, Tiangang, and Sergey Dolgov. 2022. Deep Composition of Tensor-Trains Using Squared Inverse Rosenblatt Transports.” Foundations of Computational Mathematics 22 (6): 1863–1922.
Marzouk, Youssef, Tarek Moselhy, Matthew Parno, and Alessio Spantini. 2016. Sampling via Measure Transport: An Introduction.” In Handbook of Uncertainty Quantification, edited by Roger Ghanem, David Higdon, and Houman Owhadi, 1:1–41. Cham: Springer Heidelberg.
Panaretos, Victor M., and Yoav Zemel. 2019. Statistical Aspects of Wasserstein Distances.” Annual Review of Statistics and Its Application 6 (1): 405–31.
Parno, Matthew D., and Youssef M. Marzouk. 2018. Transport Map Accelerated Markov Chain Monte Carlo.” SIAM/ASA Journal on Uncertainty Quantification 6 (2): 645–82.
Perrot, Michaël, Nicolas Courty, Rémi Flamary, and Amaury Habrard. n.d. “Mapping Estimation for Discrete Optimal Transport,” 9.
Spantini, Alessio. 2017. On the low-dimensional structure of Bayesian inference.” Thesis, Massachusetts Institute of Technology.
Spantini, Alessio, Ricardo Baptista, and Youssef Marzouk. 2022. Coupling Techniques for Nonlinear Ensemble Filtering.” SIAM Review 64 (4): 921–53.
Spantini, Alessio, Daniele Bigoni, and Youssef Marzouk. 2017. Inference via Low-Dimensional Couplings.” Journal of Machine Learning Research 19 (66): 2639–709.
Zahm, Olivier, Paul Constantine, Clémentine Prieur, and Youssef Marzouk. 2018. Gradient-Based Dimension Reduction of Multivariate Vector-Valued Functions.” arXiv:1801.07922 [Math], January.
Zhao, Yiran, and Tiangang Cui. 2023. Tensor-Based Methods for Sequential State and Parameter Estimation in State Space Models.” arXiv.

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