Efficient and Stable Multi-Dimensional Kolmogorov-Smirnov Distance

Peter Matthew Jacobs; Foad Namjoo; Jeff M. Phillips

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arXiv · 2504.11299

Efficient and Stable Multi-Dimensional Kolmogorov-Smirnov Distance

Peter Matthew Jacobs·Foad Namjoo·Jeff M. Phillips

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Abstract

We revisit extending the Kolmogorov-Smirnov distance between probability distributions to the multi-dimensional setting, and make new arguments about the proper way to approach this generalization. Our proposed formulation maximizes the difference over orthogonal dominating rectangular ranges (d-sided rectangles in R^d), and is an integral probability metric. We also prove that the distance between a distribution and a sample from the distribution converges to 0 as the sample size grows, and bound this rate. Moreover, we show that one can, up to this same approximation error, compute the distance efficiently in 4 or fewer dimensions; specifically, the runtime is near-linear in the size of the sample needed for that error. With this, we derive a delta-precision two-sample hypothesis test using this distance. Finally, we show these metrics and approximation properties do not hold for other popular variants.

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BibTeX RIS

Peter Matthew Jacobs, Foad Namjoo, Jeff M. Phillips. 2025-04-15. Efficient and Stable Multi-Dimensional Kolmogorov-Smirnov Distance. https://arxiv.org/abs/2504.11299

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