# Magnitude Distance: A Geometric Measure of Dataset Similarity

arXiv: https://arxiv.org/abs/2602.08859 · HTML: https://mlsanity.com/paper/2602.08859/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 41 references that are in the map.
- First posted on arXiv 9 Feb 2026.

## Builds on (most cited first, 14 of 14)

- [Improved Techniques for Training GANs](https://mlsanity.com/paper/1606.03498/) (2016)
- [Improved Training of Wasserstein GANs](https://mlsanity.com/paper/1704.00028/) (2017)
- [Improving and generalizing flow-based generative models with minibatch optimal transport](https://mlsanity.com/paper/2302.00482/) (2023)
- [MMD GAN: Towards Deeper Understanding of Moment Matching Network](https://mlsanity.com/paper/1705.08584/) (2017)
- [Statistical Aspects of Wasserstein Distances](https://mlsanity.com/paper/1806.05500/) (2018)
- [Inductive Moment Matching](https://mlsanity.com/paper/2503.07565/) (2025)
- [Generative Modeling using the Sliced Wasserstein Distance](https://mlsanity.com/paper/1803.11188/) (2018)
- [MMD Aggregated Two-Sample Test](https://mlsanity.com/paper/2110.15073/) (2021)
- [Can Push-forward Generative Models Fit Multimodal Distributions?](https://mlsanity.com/paper/2206.14476/) (2022)
- [A Test of Relative Similarity For Model Selection in Generative Models](https://mlsanity.com/paper/1511.04581/) (2015)
- [Metric Space Magnitude for Evaluating the Diversity of Latent Representations](https://mlsanity.com/paper/2311.16054/) (2023)
- [Metric Space Magnitude and Generalisation in Neural Networks](https://mlsanity.com/paper/2305.05611/) (2023)
- [Topological Generalization Bounds for Discrete-Time Stochastic Optimization Algorithms](https://mlsanity.com/paper/2407.08723/) (2024)
- [Approximating Metric Magnitude of Point Sets](https://mlsanity.com/paper/2409.04411/) (2024)

## Built on by (most cited first, 2 of 2)

- [Magnitude-Based Features for Multispecies Spatial Data](https://mlsanity.com/paper/2606.11775/) (2026)
- [Generalization in Neural Networks Through the Lens of Magnitude Potential](https://mlsanity.com/paper/2610.01633/) (2026)

## Abstract (arXiv)

Quantifying the distance between datasets is a fundamental question in mathematics and machine learning. We propose textitmagnitude 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.