Quantifying the distance between datasets is a fundamental question in mathematics and machine learning. We propose \textit{magnitude distance}, a novel distance metric defined on finite datasets using the notion of the \emph{magnitude} of a metric space. The proposed distance incorporates a tunable scaling parameter, $t$, that controls the sensitivity to global structure (small $t$) and finer details (large $t$). We prove several theoretical properties of magnitude distance, including its limiting behavior across scales and conditions under which it satisfies key metric properties. In contrast to classical distances, we show that magnitude distance remains discriminative in high-dimensional settings when the scale is appropriately tuned. We further demonstrate how magnitude distance can be used as a training objective for push-forward generative models. Our experimental results support our theoretical analysis and demonstrate that magnitude distance provides meaningful signals, comparable to established distance-based generative approaches.

翻译：量化数据集之间的距离是数学与机器学习中的一个基本问题。本文提出\textit{幅度距离}，这是一种基于度量空间\textit{幅度}概念、定义在有限数据集上的新型距离度量。所提出的距离包含一个可调尺度参数$t$，该参数控制对全局结构（小$t$值）与精细细节（大$t$值）的敏感度。我们证明了幅度距离的若干理论性质，包括其在不同尺度下的极限行为以及满足关键度量性质的条件。与经典距离度量相比，我们表明当尺度经过适当调整时，幅度距离在高维场景中仍能保持判别力。我们进一步展示了如何将幅度距离用作前推生成模型的训练目标。实验结果支持了我们的理论分析，并表明幅度距离能够提供有意义的信号，其效果可与基于距离的成熟生成方法相媲美。