End-to-end consistency of gBottomup

Establish a full end-to-end consistency result for gBottomup that covers the Force merge stage and the data-dependent candidate set passed to Phase 3, together with any additional assumptions required for such an analysis.

Background

Proposition 1 proves a local protection property: under uniform approximation and separation conditions, representative boundaries associated with true change-points are not removed by ordinary merges in Phase 1 or the ordinary part of Phase 2 with probability tending to one. However, the result does not cover the terminal Force merge stage or the data-dependent candidate set subsequently supplied to Phase 3 for model selection. The paper therefore identifies a complete consistency analysis, including these components, as unresolved and notes that additional assumptions would be needed.

References

A full end-to-end consistency analysis covering Force merge and the data-dependent candidate set passed to Phase 3 would require additional assumptions and is left for future work.

— Multiple change-point detection via bottom-up scanning  (2609.31116 - Park et al., 25 Sep 2026) in Section 2.4, Properties of the algorithm, immediately after Proposition 1 (Local protection property of the Unmerge rule)

It is therefore natural to investigate whether a sharper analysis can establish $m=\mathcal O_{\mathbb P}(1)$ for algorithmic outputs under suitable distributional and occupancy assumptions. This remains a conjecture: averages from finite-dataset subsampling cannot establish an asymptotic rate or reliably distinguish constant behavior from slow logarithmic growth.

— Two-Point Local Optimality in $k$-Means via Boundary-Point Screening  (2610.06182 - Lyu et al., 5 Oct 2026) in Appendix, Section “Supplementary Experimental Discussion,” subsection “Candidate-Set Scaling Protocol and Interpretation”