Transport maps

Inference by measure transport, low-dimensional coupling…

April 4, 2018 — February 21, 2023

approximation

Bayes

density

likelihood free

nonparametric

optimization

probabilistic algorithms

probability

statistics

Maps moving one probability measure into another. Most commonly seen in normalising flows. 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.

1 References

Bonnotte. 2012. “From Knothe’s Rearrangement to Brenier’s Optimal Transport Map.”

Carlier, Galichon, and Santambrogio. 2008. “From Knothe’s Transport to Brenier’s Map and a Continuation Method for Optimal Transport.”

Cui, and Dolgov. 2022. “Deep Composition of Tensor-Trains Using Squared Inverse Rosenblatt Transports.” Foundations of Computational Mathematics.

Cunningham, Zabounidis, Agrawal, et al. 2020. “Normalizing Flows Across Dimensions.”

Marzouk, Moselhy, Parno, et al. 2016. “Sampling via Measure Transport: An Introduction.” In Handbook of Uncertainty Quantification.

Panaretos, and Zemel. 2019. “Statistical Aspects of Wasserstein Distances.” Annual Review of Statistics and Its Application.

Parno, and Marzouk. 2018. “Transport Map Accelerated Markov Chain Monte Carlo.” SIAM/ASA Journal on Uncertainty Quantification.

Perrot, Courty, Flamary, et al. n.d. “Mapping Estimation for Discrete Optimal Transport.”

Spantini. 2017. “On the low-dimensional structure of Bayesian inference.”

Spantini, Baptista, and Marzouk. 2022. “Coupling Techniques for Nonlinear Ensemble Filtering.” SIAM Review.

Spantini, Bigoni, and Marzouk. 2017. “Inference via Low-Dimensional Couplings.” Journal of Machine Learning Research.

Zahm, Constantine, Prieur, et al. 2018. “Gradient-Based Dimension Reduction of Multivariate Vector-Valued Functions.” arXiv:1801.07922 [Math].

Zhao, and Cui. 2023. “Tensor-Based Methods for Sequential State and Parameter Estimation in State Space Models.”