# Field Codes for Distributed Coupling Samplers and Certified Empirical Transport

arXiv: https://arxiv.org/abs/2607.27078 · HTML: https://mlsanity.com/paper/2607.27078/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 22 of its 33 references that are in the map.
- First posted on arXiv 29 Jul 2026.

## Builds on (most cited first, 22 of 22)

- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow](https://mlsanity.com/paper/2209.03003/) (2022)
- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Improving and generalizing flow-based generative models with minibatch optimal transport](https://mlsanity.com/paper/2302.00482/) (2023)
- [Computational Optimal Transport](https://mlsanity.com/paper/1803.00567/) (2018)
- [Near-linear time approximation algorithms for optimal transport via Sinkhorn iteration](https://mlsanity.com/paper/1705.09634/) (2017)
- [Simulation-free Schr"odinger bridges via score and flow matching](https://mlsanity.com/paper/2307.03672/) (2023)
- [Variational Rectified Flow Matching](https://mlsanity.com/paper/2502.09616/) (2025)
- [On Projection Robust Optimal Transport: Sample Complexity and Model Misspecification](https://mlsanity.com/paper/2006.12301/) (2020)
- [Fast Approximation of the Sliced-Wasserstein Distance Using Concentration of Random Projections](https://mlsanity.com/paper/2106.15427/) (2021)
- [Decentralize and Randomize: Faster Algorithm for Wasserstein Barycenters](https://mlsanity.com/paper/1806.03915/) (2018)
- [DOTmark - A Benchmark for Discrete Optimal Transport](https://mlsanity.com/paper/1610.03368/) (2016)
- [Monge, Bregman and Occam: Interpretable Optimal Transport in High-Dimensions with Feature-Sparse Maps](https://mlsanity.com/paper/2302.04065/) (2023)
- [Sliced Wasserstein with Random-Path Projecting Directions](https://mlsanity.com/paper/2401.15889/) (2024)
- [Parallel Streaming Wasserstein Barycenters](https://mlsanity.com/paper/1705.07443/) (2017)
- [VQ-Flows: Vector Quantized Local Normalizing Flows](https://mlsanity.com/paper/2203.11556/) (2022)
- [Distributed Computation of Wasserstein Barycenters over Networks](https://mlsanity.com/paper/1803.02933/) (2018)
- [New Streaming Algorithms for High Dimensional EMD and MST](https://mlsanity.com/paper/2111.03528/) (2021)
- [Distributed Wasserstein Barycenters via Displacement Interpolation](https://mlsanity.com/paper/2012.08610/) (2020)
- [Communication complexity of approximate maximum matching in the message-passing model](https://mlsanity.com/paper/1704.08462/) (2017)
- [Distributed Online Optimization for Multi-Agent Optimal Transport](https://mlsanity.com/paper/1804.01572/) (2018)
- [Federated Sinkhorn](https://mlsanity.com/paper/2502.07021/) (2025)

## Abstract (arXiv)

In this paper, we formulate three communication tasks for empirical optimal transport: distributed coupling sampling, cost-evaluable coupling output, and scalar value-certified sampling. Our main result is a field-code compiler: any communicated transport field approximating an optimal empirical Monge map to error η can be completed by sparse target-cell residuals into an exact-marginal value-certified sampler with scalar certificate W1(μ,nu)≤ U≤ W1(μ,nu)+2Delta, where Delta is the public target-partition diameter. The certificate accuracy is controlled by Delta alone. The field error η controls residual communication under a cell-margin condition; without a margin, η alone does not bound residuals. We instantiate the compiler via adaptive local-affine and tensor-product spline codes with d(m+1)db field bits in the spline case, plus residual lists charged separately. For lower bounds, exact Gap-Hamming embeddings prove certified output is hard, including a smooth cell-packing diffeomorphism family requiring Omega(varepsilon-2d/(d+4)) communication for any cost-evaluable, cost-certified, or value-certified protocol. The same gadgets admit zero-communication samplers, formally separating the sampler and certificate-bearing output models. These results identify the transport field as the right communicated object whenever a field code is available, primarily as a residual-sparsity tool.