Sliced Wasserstein Sketches with Error Certificates

Abstract:Compact representations that support geometrically meaningful comparisons are valuable for learning from probability distributions. We study optimal, free-weight $M$-atom quantization of one-dimensional probability measures in quadratic Wasserstein distance and extend it to higher dimensions via slicing. Using $L$ shared projections yields a sketch with $O(LM)$ comparison time. Our main result is an exact decomposition of squared distance error into residual and partition-mismatch terms, yielding computable certificates for arbitrary centroid sketches. For optimal sketches, a finite $(2+\delta)$ moment for some $\delta>0$ guarantees $O(M^{-1})$ error in the finite-direction sliced distance, while equal-mass sketches can converge more slowly. Each slice preserves its mean, while storing residual energies enables covariance recovery from sufficiently many generic directions. Experiments on document, point-cloud, and flow-cytometry benchmarks show that optimal sketches substantially reduce storage while preserving the classification performance of uncompressed sliced distances. They also improve generalization from few labelled subjects and support end-to-end learning as compact pooling layers.

Machine Learning (cs.LG)

arXiv:2610.20788 [cs.LG]

(or arXiv:2610.20788v1 [cs.LG] for this version)

https://doi.org/10.48550/arXiv.2610.20788 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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