MST-Direct at Scale: Multivariate and Conditional Geostatistical Simulation via Sinkhorn Optimal Transport
This paper extends MST-Direct, a Sinkhorn optimal transport approach, to multivariate, conditional, and large-scale geostatistical simulation. It addresses scalability via a sparse candidate-restricted Sinkhorn matcher with O(nC) memory, extends to multiple variables using an FFT-MA Gaussian backbone, and enables hard-data conditioning through kriging. Validation on a six-variate heteroscedastic distribution shows zero histogram error and exact spatial correlation reproduction, outperforming the PPMT approximation.
[2606.07578] MST-Direct at Scale: Multivariate and Conditional Geostatistical Simulation via Sinkhorn Optimal Transport
[Submitted on 26 May 2026]
Title:MST-Direct at Scale: Multivariate and Conditional Geostatistical Simulation via Sinkhorn Optimal Transport
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Abstract:This paper extends MST-Direct, a Matching-via-Sinkhorn-Transport approach for multivariate geostatistical simulation, from the original bivariate, unconditional, small-grid formulation to multivariate, conditional, and large-grid settings. We address the three main limitations identified in the original work: (i) scalability beyond a few thousand nodes through a sparse, candidate-restricted Sinkhorn matcher with O(nC) memory complexity; (ii) extension to multiple variables by matching target value tuples onto an independent FFT-MA Gaussian backbone that reproduces a prescribed variogram; and (iii) hard-data conditioning by fixing observed data tuples at their spatial locations while conditioning the backbone through kriging. Because the transport plan remains a permutation of the target tuples, the multivariate joint distribution is preserved exactly.
The method is validated using the same six-variate, heteroscedastic, strongly nonlinear reference distribution employed in Direct Multivariate Simulation (DMS), under both unconditional (200x200) and conditional (100x100, 200 hard-data samples) scenarios, and is benchmarked against the Projection Pursuit Multivariate Transform (PPMT). Results show that MST-Direct reproduces the joint distribution with zero histogram error, exactly honours hard data, and accurately reproduces the prescribed spatial correlation structure, whereas PPMT remains an approximation.
Index Terms-Optimal transport, Sinkhorn algorithm, geostatistical simulation, multivariate simulation.
Subjects:
Machine Learning (cs.LG); Methodology (stat.ME); Machine Learning (stat.ML)
Cite as: arXiv:2606.07578 [cs.LG]
(or arXiv:2606.07578v1 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2606.07578
arXiv-issued DOI via DataCite
Submission history
From: Tcharlies Schmitz [view email] [v1] Tue, 26 May 2026 20:20:06 UTC (2,274 KB)
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