Verify the common-basis assumption induced by adjustment-vector clustering

Determine whether clusters produced by the Stage-2 adjustment-vector clustering procedure satisfy the common-objective and common non-degenerate optimal-basis assumption required for the direction-dispersion bound.

Background

The theoretical bound relating cluster direction heterogeneity to within-cluster adjustment-vector variance requires all clients in a cluster to share objective functions and a common non-degenerate optimal basis. The clustering procedure groups clients using bottleneck orderings and adjustment vectors, but the paper does not establish that these groups satisfy the common-basis condition. Resolving this issue would connect the practical clustering rule to the quantitative convergence bound rather than treating direction heterogeneity as an independently measured quantity.

References

Whether a cluster produced by the Stage-2 procedure satisfies this assumption is not verified by the procedure itself; the bound equation 10 below is therefore a statement about this restricted case, and outside it HCn is a measured quantity.

— Collaborative Personalized Preference Alignment for LLMs under Data Deficiency  (2610.05898 - Yang et al., 5 Oct 2026) in Remark H.8, Appendix H.2, p. 21