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 $(2+δ)$ moment for some $δ>0$ 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.