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Adapted optimal transport between Gaussian processes in discrete time
Published 9 Apr 2024 in math.PR | (2404.06625v4)
Abstract: We derive explicitly the adapted $2$-Wasserstein distance between non-degenerate Gaussian distributions on $\mathbb{R}N$ and characterize the optimal bicausal coupling(s). This leads to an adapted version of the Bures-Wasserstein distance on the space of positive definite matrices.
- M. Agueh and G. Carlier. Barycenters in the Wasserstein space. SIAM Journal on Mathematical Analysis, 43(2):904–924, 2011.
- Causal discovery via monotone triangular transport maps. In NeurIPS 2023 Workshop Optimal Transport and Machine Learning, 2023.
- All adapted topologies are equal. Probability Theory and Related Fields, 178:1125–1172, 2020.
- Estimating processes in adapted Wasserstein distance. The Annals of Applied Probability, 32(1):529–550, 2022.
- Fundamental properties of process distances. Stochastic Processes and their Applications, 130(9):5575–5591, 2020.
- Causal transport in discrete time and applications. SIAM Journal on Optimization, 27(4):2528–2562, 2017.
- Adapted Wasserstein distance between the laws of SDEs. arXiv preprint arXiv:2209.03243, 2022.
- On the representation and learning of monotone triangular transport maps. Foundations of Computational Mathematics, pages 1–46, 2023.
- The Wasserstein space of stochastic processes. arXiv preprint arXiv:2104.14245, 2021.
- The Knothe-Rosenblatt distance and its induced topology. arXiv preprint arXiv:2312.16515, 2023.
- On the Bures–Wasserstein distance between positive definite matrices. Expositiones Mathematicae, 37(2):165–191, 2019.
- J. Bion-Nadal and D. Talay. On a Wasserstein-type distance between solutions to stochastic differential equations. The Annals of Applied Probability, 29(3):1609–1639, 2019.
- On the theory of elliptically contoured distributions. Journal of Multivariate Analysis, 11(3):368–385, 1981.
- S. Eckstein and G. Pammer. Computational methods for adapted optimal transport. The Annals of Applied Probability, 34(1A):675–713, 2024.
- M. Gelbrich. On a formula for the L2superscript𝐿2L^{2}italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT Wasserstein metric between measures on Euclidean and Hilbert spaces. Mathematische Nachrichten, 147(1):185–203, 1990.
- B. Han. Distributionally robust Kalman filtering with volatility uncertainty. arXiv preprint arXiv:2302.05993, 2023.
- Entropic optimal transport between unbalanced Gaussian measures has a closed form. Advances in Neural Information Processing Systems, 33:10468–10479, 2020.
- R. Lassalle. Causal transport plans and their Monge–Kantorovich problems. Stochastic Analysis and Applications, 36(3):452–484, 2018.
- Z. Lin. Riemannian geometry of symmetric positive definite matrices via Cholesky decomposition. SIAM Journal on Matrix Analysis and Applications, 40(4):1353–1370, 2019.
- Wasserstein Riemannian geometry of Gaussian densities. Information Geometry, 1:137–179, 2018.
- R. J. McCann. A convexity principle for interacting gases. Advances in Mathematics, 128(1):153–179, 1997.
- R. J. Muirhead. Aspects of Multivariate Statistical Theory. John Wiley & Sons, 1982.
- F. Otto. The geometry of dissipative evolution equations: the porous medium equation. Communications in Partial Differential Equations, 26:101–174, 2001.
- G. Pammer. A note on the adapted weak topology in discrete time. Electronic Communications in Probability, 29:1–13, 2024.
- B. A. Robinson and M. Szölgyenyi. Bicausal optimal transport for SDEs with irregular coefficients. arXiv preprint arXiv:2403.09941, 2024.
- A. Takatsu. Wasserstein geometry of Gaussian measures. Osaka Journal of Mathematics, 48(4):1005–1026, 2011.
- M. Zorzi. Optimal transport between Gaussian stationary processes. IEEE Transactions on Automatic Control, 66(10):4939–4944, 2020.
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