Copula Transformations for Data-Consistent Inversion

Data-consistent inversion (DCI) constructs probability measures whose push-forward distributions agree with observed data, while iterative data-consistent inversion (iDCI) extends this framework to generalized stochastic inverse problems by enforcing multiple push-forward constraints sequentially. Although iDCI avoids the direct approximation of high-dimensional joint densities, its relationship to the original joint DCI solution has remained unclear. In this work, we establish this relationship through copula theory. Using Sklar's theorem, we derive a factorization of the DCI update into separate marginal and dependence transformations and show that the discrepancy remaining after convergence of the iDCI algorithm is entirely characterized by the copulas associated with the observed and predicted joint distributions. This characterization motivates a copula-transformed iDCI solution, and we prove that an exact copula transformation recovers the original DCI solution. We further establish convergence results for approximate copula transformations under converging sequences of reference measures and progressively enriched feasible sets. Numerical examples demonstrate how the geometry induced by the quantity-of-interest map governs the importance of the copula transformation, illustrate an adaptive reference-measure refinement strategy for improving computational accuracy under a fixed sampling budget, and demonstrate the progressive refinement of generalized stochastic inverse problems through heterogeneous, asynchronously acquired experiments.

Wasserstein gradient flows for Coulomb discrepancies

We study the long-time behavior of the Wasserstein gradient flow of the squared Maximum Mean Discrepancy (MMD) between a probability measure $ρ$ and a target measure $μ$, where the underlying kernel is given by a Coulomb potential. For $L^\infty$ target densities $μ$, we establish the existence of global weak solutions starting from arbitrary Borel probability measures and prove that the density $ρ_t$ belongs to $L^\infty$ for any $t>0$. We also show that the Hölder norm can grow exponentially in time. On the flat torus ${\mathbb{T}}^\mathsf{d}$, we prove a global metric PL inequality for every finite-Coulomb-energy source and nearly uniform target. For general bounded, uniformly positive targets, we prove exponential decay of the squared MMD without requiring a lower bound on the initial data, using a defective PL inequality. We also prove that the usual PL inequality may fail when the target vanishes only at one point and that, when $\mathsf{d}\ge2$, no PL constant can hold uniformly over all targets satisfying a prescribed lower bound. On ${\mathbb{R}}^\mathsf{d}$, for $\mathsf{d}\ge2$, under radial symmetry, source-support inclusion, and target-positivity assumptions, we establish a PL inequality and exponential convergence. On the unrestricted whole-space class, neither a multiplicative squared-MMD decay modulus uniform over the initial datum nor a global PL inequality can hold. Finally, in every dimension and in both spatial settings, we prove that every Lagrangian critical point coincides with the target when $(ρ-μ)^+$ is absolutely continuous. In dimension two, the energy supplies uniform tightness. This implies that if our constructed solutions have finite energy at some positive time, then they converge to the target narrowly and strongly in negative-order Sobolev spaces.

Another Look at Log-PCA for Probability Measures: A Dynamical Formulation and Statistical Convergence

This paper is concerned with learning principal variations of random probability measures on $\mathbb{R}^m$ under the Wasserstein geometry. We introduce a new dynamical formulation to interpret the log-PCA, a linearized principal geodesic analysis, as a variational approach. Our differentiable version, termed as the Wasserstein Tangential PCA (WT-PCA), captures the local principal modes of geodesic variations of a (weighted) probability measure on the Wasserstein space via its covariance operator at barycenter. Based on the dynamical perspective and leveraging parallel transport structure of the optimal transport problems, we derive a general statistical convergence rate of the empirical WT-PCA when estimated from data in terms of the 2-Wasserstein distance between the population and empirical barycenter reference measures.

Identifiability and Stability of Generative Drifting with Companion-Elliptic Kernel Families

This paper analyzes identifiability and stability for the drifting field underlying distributional matching in the Generative Drifting framework of Deng et al. First, we introduce the class of companion-elliptic kernels, which includes the Laplace kernel and is characterized by a second-order elliptic coupling between each kernel $κ$ in this class and its companion function $η$. For each kernel in this class and each pair of Borel probability measures, we prove that the drifting field vanishes if and only if the two probability measures are equal. We further show that this class consists precisely of Gaussian kernels and Matérn kernels with $ν\ge 1/2$. Second, by constructing counterexamples, we exhibit sequences for which mass escapes to infinity while the field tends to zero; in particular, control of the field norm alone does not guarantee weak convergence. Nevertheless, we prove that the only possible mode of failure is confined to the one-dimensional ray $\{c\,p:0\le c\le 1\}$. Consequently, weak convergence can be restored by imposing an asymptotic lower bound on the intrinsic overlap scalar, a linear observable defined by the kernel and the target measure.