深度重心回归用于最优传输映射估计及其统计最优性

中文摘要 AI 辅助

最优传输（OT）映射提供了一种用于对齐概率分布的几何变换，并已成为机器学习中的有用工具。然而，现有的OT映射估计器在尖锐的统计保证与基于稳定训练目标的实用参数化估计之间仍存在差距。理论估计器达到极小极大最优收敛速率，但它们通常是非参数的，且可能带来要求高的实现设计或推断成本。实用估计器是参数化且可扩展的，但其统计保证仍未被充分探索，且其极小极大、对抗式训练目标可能对优化算法敏感。我们提出BROT（用于OT的重心回归），一种简单的两步方法，首先计算无正则化的OT计划，然后通过最小二乘回归将深度神经网络（DNN）拟合到诱导的重心目标。在标准正则性条件下，我们证明当真实OT映射是Lipschitz时，BROT的DNN估计器达到极小极大收敛速率。在合成数据集和图像数据集上的数值研究表明，与现有估计方法相比，BROT提供了准确的映射估计、强目标分布匹配和有竞争力的传输成本。在两个下游任务（单细胞扰动预测和无监督域适应）上的实验进一步表明，BROT的准确估计可以转化为更强的任务性能。

英文摘要

The optimal transport (OT) map provides a geometric transformation for aligning probability distributions and has become a useful tool in machine learning. However, existing estimators of the OT map still exhibit a gap between sharp statistical guarantees and practical parametric estimation based on stable training objectives. Theoretical estimators achieve minimax optimal convergence rates, but they are typically nonparametric and can incur demanding implementation design or inference costs. Practical estimators are parametric and scalable, but their statistical guarantees remain underexplored, and their min-max, adversarial-like training objectives can be sensitive to optimization algorithms. We propose BROT (Barycentric Regression for OT), a simple two-step method that first computes the unregularized OT plan and then fits a deep neural network (DNN) to the induced barycentric targets by least-squares regression. Under standard regularity conditions, we prove that the DNN estimator of BROT attains the minimax convergence rate, when the ground-truth OT map is Lipschitz. Numerical studies on synthetic datasets and an image dataset show that BROT provides accurate map estimates, strong target distribution matching, and competitive transport costs, compared to existing estimation methods. Experiments on two downstream tasks, single-cell perturbation prediction and unsupervised domain adaptation, further suggest that the accurate estimation of BROT can translate into stronger task performance.

发表机构

• KAIST AI（韩国科学技术院人工智能学院）
• UNIST（蔚山科学技术院）
• Seoul National University（首尔大学）

机构由 AI 辅助整理，请以论文原文为准。