A1350
Title: Wasserstein transport between copulas via a Benamou-Brenier-type dynamical formulation
Authors: Na Luo - Karlsruhe Institute of Technology (Germany) [presenting]
Oliver Grothe - Karlsruhe Institute of Technology (Germany)
Abstract: Optimal interpolations between copulas, i.e., continuously transforming one copula into another, via Benamou-Brenier mass transport under the constraint that uniform marginals are preserved along the entire interpolation path, are studied. However, restricting the Benamou-Brenier formulation to the copula domain leads to an intractable global constraint, since mass transport simultaneously affects both dependence structures and marginal distributions. To overcome this difficulty, a structural correspondence between dependence and marginal flows is established, termed a Sklar-type decomposition for flows. This yields a tractable reformulation of the constrained problem and allows construction of admissible interpolation paths of copulas and the associated vector fields. As a consequence, a transport-based distance on the space of copulas is defined as the minimal action required to transform one copula into another within the copula domain. This distance defines a metric, termed the copula-constrained Wasserstein distance. The approach is illustrated using neural network parameterizations and interpolations between different dependence structures are demonstrated. More generally, the results enable interpolation between multivariate distributions under fixed marginal constraints, demonstrated with an example involving Gaussian marginals.
