Regularity of Solutions to Beckmann's Parametric Optimal Transport

• URL: http://arxiv.org/abs/2603.19755v1
• Date: Fri, 20 Mar 2026 08:40:37 GMT
• Title: Regularity of Solutions to Beckmann's Parametric Optimal Transport
• Abstract summary: We show that the Lagrangian multiplier that enforces Beckmann's divergence constraint fulfills a Poisson equation.<n>The exact Hlder regularity of the potential, the flux and the flow generating is derived on the basis of Hlder regularity of source and target densities.<n>We show that this approach generalizes to other probability paths, like Fisher-Rao gradient flows.
• Score: 2.915947686247852
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Beckmann's problem in optimal transport minimizes the total squared flux in a continuous transport problem from a source to a target distribution. In this article, the regularity theory for solutions to Beckmann's problem in optimal transport is developed utilizing an unconstrained Lagrangian formulation and solving the variational first order optimality conditions. It turns out that the Lagrangian multiplier that enforces Beckmann's divergence constraint fulfills a Poisson equation and the flux vector field is obtained as the potential's gradient. Utilizing Schauder estimates from elliptic regularity theory, the exact Hölder regularity of the potential, the flux and the flow generating is derived on the basis of Hölder regularity of source and target densities on a bounded, regular domain. If the target distribution depends on parameters, as is the case in conditional (``promptable'') generative learning, we provide sufficient conditions for separate and joint Hölder continuity of the resulting vector field in the parameter and the data dimension. Following a recent result by Belomnestny et al., one can thus approximate such vector fields with deep ReQu neural networks in C^(k,alpha)-Hölder norm. We also show that this approach generalizes to other probability paths, like Fisher-Rao gradient flows.

Related papers

• Neural Optimal Transport Meets Multivariate Conformal Prediction [58.43397908730771]
  We propose a framework for conditional vectorile regression (CVQR)<n>CVQR combines neural optimal transport with quantized optimization, and apply it to predictions.
  arXiv  Detail & Related papers  (2025-09-29T19:50:19Z)
• Hessian-guided Perturbed Wasserstein Gradient Flows for Escaping Saddle Points [54.06226763868876]
  Wasserstein flow (WGF) is a common method to perform optimization over the space of measures.<n>We show that PWGF converges to a global optimum in terms of general non objectives.
  arXiv  Detail & Related papers  (2025-09-21T08:14:20Z)
• Constrained Gaussian Wasserstein Optimal Transport with Commutative Covariance Matrices [24.878352716406418]
  We consider three types of constraints, motivated by perception-aware lossy compression, generative principal component analysis, and deep joint source-channel coding.<n>We derive explicit results for the minimum achievable mean squared error under the three aforementioned constraints.
  arXiv  Detail & Related papers  (2025-03-05T18:56:48Z)
• Flow-based Distributionally Robust Optimization [23.232731771848883]
  We present a framework, called $textttFlowDRO$, for solving flow-based distributionally robust optimization (DRO) problems with Wasserstein uncertainty sets. We aim to find continuous worst-case distribution (also called the Least Favorable Distribution, LFD) and sample from it. We demonstrate its usage in adversarial learning, distributionally robust hypothesis testing, and a new mechanism for data-driven distribution perturbation differential privacy.
  arXiv  Detail & Related papers  (2023-10-30T03:53:31Z)
• Resampling Base Distributions of Normalizing Flows [0.0]
  We introduce a base distribution for normalizing flows based on learned rejection sampling. We develop suitable learning algorithms using both maximizing the log-likelihood and the optimization of the reverse Kullback-Leibler divergence.
  arXiv  Detail & Related papers  (2021-10-29T14:44:44Z)
• Optimal policy evaluation using kernel-based temporal difference methods [78.83926562536791]
  We use kernel Hilbert spaces for estimating the value function of an infinite-horizon discounted Markov reward process. We derive a non-asymptotic upper bound on the error with explicit dependence on the eigenvalues of the associated kernel operator. We prove minimax lower bounds over sub-classes of MRPs.
  arXiv  Detail & Related papers  (2021-09-24T14:48:20Z)
• Variational Transport: A Convergent Particle-BasedAlgorithm for Distributional Optimization [106.70006655990176]
  A distributional optimization problem arises widely in machine learning and statistics. We propose a novel particle-based algorithm, dubbed as variational transport, which approximately performs Wasserstein gradient descent. We prove that when the objective function satisfies a functional version of the Polyak-Lojasiewicz (PL) (Polyak, 1963) and smoothness conditions, variational transport converges linearly.
  arXiv  Detail & Related papers  (2020-12-21T18:33:13Z)
• Direct Optimal Control Approach to Laser-Driven Quantum Particle Dynamics [77.34726150561087]
  We propose direct optimal control as a robust and flexible alternative to indirect control theory. The method is illustrated for the case of laser-driven wavepacket dynamics in a bistable potential.
  arXiv  Detail & Related papers  (2020-10-08T07:59:29Z)
• Bayesian ODE Solvers: The Maximum A Posteriori Estimate [30.767328732475956]
  It has been established that the numerical solution of ordinary differential equations can be posed as a nonlinear Bayesian inference problem. The maximum a posteriori estimate corresponds to an optimal interpolant in the Hilbert space associated with the prior. The methodology developed provides a novel and more natural approach to study the convergence of these estimators.
  arXiv  Detail & Related papers  (2020-04-01T11:39:59Z)
• A Near-Optimal Gradient Flow for Learning Neural Energy-Based Models [93.24030378630175]
  We propose a novel numerical scheme to optimize the gradient flows for learning energy-based models (EBMs) We derive a second-order Wasserstein gradient flow of the global relative entropy from Fokker-Planck equation. Compared with existing schemes, Wasserstein gradient flow is a smoother and near-optimal numerical scheme to approximate real data densities.
  arXiv  Detail & Related papers  (2019-10-31T02:26:20Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.