发表机构
B0Labs, N2TP Technology; National Economics University; Nanyang Technological University; Phenikaa University（N2TP科技B0实验室; 国民经济大学; 南洋理工大学; Phenikaa大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出场码编译器，针对经验最优传输的三类通信任务，实现标量可验证采样，通过Gap-Hamming嵌入证明可验证输出的通信下界，区分采样器与证书输出模型，以传输场作为残差稀疏性工具。
AI 中文摘要
本文针对经验最优传输问题，构建了三类通信任务：分布式耦合采样、可计算代价的耦合输出，以及标量可验证采样。核心成果是提出了一种场码编译器：任何近似最优经验Monge映射至误差η的传输场，均可通过稀疏目标单元残差补全为精确边际、标量证书满足W₁(μ,ν)≤U≤W₁(μ,ν)+2Δ的可验证采样器，其中Δ为公共目标划分直径，证书精度仅由Δ控制。场误差η在单元边界条件下控制残差通信；无边界时，η无法单独约束残差。本文通过自适应局部仿射与张量积样条码实例化该编译器，样条情形下场比特数为d(m+1)ᵈb，残差列表单独计费。下界方面，精确Gap-Hamming嵌入证明可验证输出具有难度，包括任意可计算代价、代价可验证或值可验证协议，对于光滑单元填充微分同胚族需Ω(ε^(-2d/(d+4)))通信量。上述构造还支持零通信采样器，形式上区分了采样器与承载证书的输出模型。这些结果表明，当存在场码时，传输场是合适的通信对象，主要作为残差稀疏性工具。
英文摘要
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 $W_1(μ,ν)\leq U\leq W_1(μ,ν)+2Δ$, where $Δ$ is the public target-partition diameter. The certificate accuracy is controlled by $Δ$ 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 $Ω(\varepsilon^{-2d/(d+4)})$ 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.
