Sampleability transport, nonlinear regularization, and the porous medium flow

TL;DR

This paper explores the Wasserstein projection for probability measures with bounded density ratios, analyzing porous medium flow and nonlinear diffusion, and establishing foundational properties and obstructions within this framework.

Contribution

It proves existence and uniqueness of the sampleability projection, analyzes porous medium equations as regularizers, and identifies structural obstructions in the nonlinear diffusion approach.

Findings

Proved existence and uniqueness of the sampleability projection.

Analyzed porous medium equation properties like finite support propagation.

Identified a structural obstruction in porous-medium sampleability theory.

Abstract

We study the Wasserstein projection of a compactly supported probability measure onto the class of measures whose density ratio is bounded, and we place this projection in a broader program connecting generative modeling, optimal transport, and nonlinear diffusion. The paper proves existence and uniqueness of the sampleability projection, uniqueness of the Brenier map at the minimizer, path independence of the quadratic Wasserstein generation loss, and the diffusion-threshold picture for the heat semigroup. The porous medium equation is then analyzed as a candidate forward regularizer. We prove the two rigorous properties that make the equation attractive for this purpose, namely finite propagation of compact support and an explicit Wasserstein cost bound obtained from dissipation of the R\'enyi entropy. We then identify a structural obstruction inherent to any porous-medium version of the sampleability theory. Every nontrivial compactly supported whole-space porous-medium profile has vanishing essential infimum on any compact set containing its support, hence infinite density ratio in the original sense, and the assertion that the porous medium flow reaches the same density-ratio sampleable class while preserving compact support is false. To isolate the mathematically valid content of the nonlinear-diffusion program, we also prove an endpoint-constrained Benamou-Brenier principle for the sampleability projection and derive the corrected spectral picture near a strictly positive equilibrium on a fixed compact domain. In that regime the leading-order damping is exponential, with quadratic mode coupling in the first nonlinear correction. The Hele-Shaw and mesa-limit interpretation is therefore presented here as a conjectural variational extension rather than as a proved theorem.