熵最优传输的最小本征维度标度
Minimum intrinsic dimension scaling for entropic optimal transport
AI总结：
本研究受流形假设启发，提出熵最优传输的最小本征维度（MID）标度现象，推导仅由单尺度本征维度最小值决定收敛速率的统计界，证明其普适性并首次严格解释熵正则化的统计效应。
摘要：
受流形假设（即外蕴维度高的数据可能具有低本征维度）的启发，我们推导了对数据本征维度敏感的熵最优传输精细统计界。我们的界涉及一种鲁棒的本征维度概念，仅在取决于正则化参数的单一距离尺度下测量，并表明正是这些单尺度本征维度的最小值决定了收敛速率。我们将此称为最小本征维度标度（MID scaling）现象，并在代价函数有界且满足Lipschitz条件、不对数据分布做任何假设的情况下建立了MID标度，且适用于除数值之外的多种熵最优传输量；当一个分布的支撑集位于流形上时，还存在更强的对应结论。我们的结果表明MID标度是一种普遍现象，显著推进了现有理论水平，并首次对熵正则化作为一种距离尺度的统计效应给出了严格解释。
英文摘要：
Motivated by the manifold hypothesis, which states that data with a high extrinsic dimension may yet have a low intrinsic dimension, we develop refined statistical bounds for entropic optimal transport that are sensitive to the intrinsic dimension of the data. Our bounds involve a robust notion of intrinsic dimension, measured at only a single distance scale depending on the regularization parameter, and show that it is only the minimum of these single-scale intrinsic dimensions which governs the rate of convergence. We call this the Minimum Intrinsic Dimension scaling (MID scaling) phenomenon, and establish MID scaling with no assumptions on the data distributions so long as the cost is bounded and Lipschitz, and for various entropic optimal transport quantities beyond just values, with stronger analogs when one distribution is supported on a manifold. Our results significantly advance the theoretical state of the art by showing that MID scaling is a generic phenomenon, and provide the first rigorous interpretation of the statistical effect of entropic regularization as a distance scale.
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