Magnitude Distance: A Geometric Measure of Dataset Similarity

Quantifying the distance between datasets is a fundamental question in mathematics and machine learning. We propose magnitude distance, a novel distance metric defined on finite datasets using the notion of the 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.