用单网络固定点优化实现更稳定的神经最优传输。
Implicit Neural Optimal Transport via Fixed-Point Optimization
- 单个势函数参数化，通过固定点求解替代对抗训练。
- 可同时恢复正向与逆向传输映射，训练更稳定且高效。
- 适合需要高精度传输和低内存的生成任务。
我们提出一种隐式神经最优传输方法，避免了现有方法中常见的对抗式极小极大优化和多网络结构。核心思想是参数化Kantorovich对偶中的单一势函数，并将相关的c-变换重构为近端固定点问题。这带来一个稳定的单网络框架，通过对偶可行性通过近端最优性条件精确强制，而非依赖对抗训练。尽管存在内部固定点计算，梯度仍可通过无需对固定点迭代求导的方式计算，实现高效训练且无需隐式微分。我们进一步建立了随机梯度下降的收敛性。所提框架高效、可扩展且应用广泛：能同时恢复前向与后向传输映射，并自然推广至类别条件设置。在高维高斯基准、物理数据集和图像翻译任务上的实验表明，该方法在传输精度、训练稳定性以及计算与内存效率方面均表现优异。
原文摘要 · Abstract (English)
We propose an implicit neural formulation of optimal transport that eliminates adversarial min--max optimization and multi-network architectures commonly used in existing approaches. Our key idea is to parameterize a single potential in the Kantorovich dual and reformulate the associated c-transform as a proximal fixed-point problem. This yields a stable single-network framework in which dual feasibility is enforced exactly through proximal optimality conditions rather than adversarial training. Despite the inner fixed-point computation, gradients can be computed without differentiating through the fixed-point iterations, enabling efficient training without requiring implicit differentiation. We further establish convergence of stochastic gradient descent. The resulting framework is efficient, scalable, and broadly applicable: it simultaneously recovers forward and backward transport maps and naturally extends to class-conditional settings. Experiments on high-dimensional Gaussian benchmarks, physical datasets, and image translation tasks demonstrate strong transport accuracy together with improved training stability and favorable computational and memory efficiency.
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