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

Tchalies Bachmann Schmitz

Abstract

Multivariate geostatistical simulation requires the faithful reproduction of complex non-linear dependencies among geological variables, including bimodal distributions, step functions, and heteroscedastic relationships. Traditional methods such as the Gaussian Copula and LU Decomposition assume linear correlation structures and often fail to preserve these complex joint distribution patterns. We propose MST-Direct (Matching via Sinkhorn Transport), a novel algorithm based on Optimal Transport theory that uses the Sinkhorn algorithm to directly match multivariate distributions while preserving spatial correlation structures. The method processes all variables simultaneously as a single multidimensional vector, enabling relational matching across the full joint space rather than relying on pairwise linear dependencies.

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

Abstract

Multivariate geostatistical simulation requires the faithful reproduction of complex non-linear dependencies among geological variables, including bimodal distributions, step functions, and heteroscedastic relationships. Traditional methods such as the Gaussian Copula and LU Decomposition assume linear correlation structures and often fail to preserve these complex joint distribution patterns. We propose MST-Direct (Matching via Sinkhorn Transport), a novel algorithm based on Optimal Transport theory that uses the Sinkhorn algorithm to directly match multivariate distributions while preserving spatial correlation structures. The method processes all variables simultaneously as a single multidimensional vector, enabling relational matching across the full joint space rather than relying on pairwise linear dependencies.
Paper Structure (30 sections, 15 equations, 3 figures, 5 tables)

This paper contains 30 sections, 15 equations, 3 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Related Work
  3. Traditional Multivariate Geostatistics
  4. Copula Methods
  5. Multivariate Transforms
  6. Optimal Transport
  7. Methodology
  8. Problem Formulation
  9. Optimal Transport Framework
  10. Entropy-Regularized Sinkhorn Algorithm
  11. Relational Matching
  12. MST-Direct Algorithm
  13. Greedy Rounding Procedure
  14. Experimental Design
  15. Synthetic Data Generation
  16. ...and 15 more sections

Figures (3)

  • Figure 1: MST-Direct algorithm pseudocode.
  • Figure 2: Scatter plot comparison showing preservation of complex correlations. Columns: Original, MST-Direct, Copula, LU. Rows: Step, Gaussian Mixture, Sinusoidal, Step Random, Heteroscedastic. MST-Direct achieves perfect shape preservation for all relationship types.
  • Figure 3: Variogram comparison for X (Spherical, range=12) and Y (Exponential, range=6). MST (red), Copula (green), LU (purple) compared against Original (black circles) and Theoretical (gray line).