MST-Direct: Matching via Sinkhorn Transport for Multivariate Geostatistical Simulation with Complex Non-Linear Dependencies

Technical Deep Dive: MST-Direct

Overview

MST-Direct is a novel algorithm for multivariate geostatistical simulation that preserves complex non-linear dependencies between variables. It applies optimal transport theory, specifically the Sinkhorn algorithm, to directly match the target joint distribution while maintaining spatial correlation structures.

Problem & Context

Multivariate geostatistical simulation is a fundamental technique for generating realistic geological models, but traditional methods like Gaussian Copula and LU Decomposition struggle to capture complex non-linear relationships commonly observed in real-world data. These include bimodal distributions, step functions, heteroscedastic patterns, and other non-linear dependencies.

Methodology

MST-Direct's key innovations are:

• Using optimal transport theory and the Sinkhorn algorithm to directly match the target multivariate distribution
• Incorporating relational matching based on k-nearest neighbor adjacency to preserve spatial coherence
• Processing all variables simultaneously as a single multidimensional vector, avoiding iterative sequential updates

The algorithm normalizes the input variables, generates random anchor points, constructs a k-nearest neighbor adjacency graph, and then performs the Sinkhorn-based relational matching to find an optimal permutation that matches the target joint distribution.

Data & Experimental Setup

The authors evaluated MST-Direct on synthetic data with five types of complex bivariate relationships:

• Step function
• Gaussian mixture
• Sinusoidal
• Random branching
• Heteroscedastic

They compared MST-Direct against two established methods: Gaussian Copula and LU Decomposition. Evaluation metrics included 2D histogram similarity for shape preservation and Pearson correlation for variogram reproduction.

Results

Key findings:

• MST-Direct achieved 100% shape preservation across all relationship types, significantly outperforming the comparison methods.
• For variable X with a spherical variogram, LU Decomposition had a slight advantage in variogram preservation.
• For variable Y with an exponential variogram, MST-Direct outperformed both alternatives, especially for the challenging Step Random case.

Interpretation

The superior shape preservation of MST-Direct is attributed to its direct matching of the target joint distribution, rather than relying on transformations to Gaussian space like the comparison methods. The relational matching component was crucial for maintaining spatial coherence.

The trade-off between shape and variogram preservation arises from the competing objectives. LU Decomposition's advantage for variogram X comes from its explicit use of the target covariance function, while MST-Direct's focus on distribution matching leads to slightly perturbed variograms in some cases.

Limitations & Uncertainties

Current limitations of MST-Direct include:

• Scalability to very large grids (>10,000 points) requires algorithmic improvements
• Extension to more than 2 variables needs careful handling of the curse of dimensionality
• Conditional simulation with hard data constraints is not yet implemented
• Anisotropic variogram support requires further development

What Comes Next

Future work will address these limitations and explore GPU acceleration for the Sinkhorn iterations. Overall, MST-Direct represents a significant advancement for geostatistical applications requiring accurate modeling of non-linear dependencies, such as mineral resource estimation, reservoir characterization, and environmental modeling.