Replication of the combinatorial variance analysis for the Minner estimator

Replicate Li et al.'s combinatorial variance-analysis approach for the Minner estimator under the circulant permutation framework, or determine precisely why that approach cannot be transferred to the estimator.

Background

The paper contrasts its sampling-without-replacement approach with the combinatorial derivation used in the original CMinHash work. The authors state that they were unable to reproduce that derivation for the Minner estimator, which leaves the transfer of the original combinatorial method unresolved.

The issue is especially relevant because the paper aims to extend circulant-permutation analysis beyond ordinary MinHash to the asymmetric Minner estimator. The remark also reports that an error in the original CMinHash variance theorem was found, further complicating direct replication of that approach.

References

We never were able to replicate Li's approach for the Minner estimator, and we thought this was because we did not understand enough combinatorics.

— Understanding Circulant Permutation to Extend the CMinHash Estimator  (2610.18551 - Kang et al., 13 Oct 2026) in Remark in Appendix A.2, immediately following Theorem 3.1 of Li et al.