Continuous relaxation from prototype weights to latent-space weighting

Derive a formal justification for the continuous relaxation connecting the cluster-level prototype weights to the latent-space weighting function, for example through local Lipschitz arguments.

Background

The paper defines prototype-level weights αk\alpha_k and a continuous latent-space relaxation w(z)w(z) to facilitate importance-weighting analysis. The bridge from prototype assignments to individual sample weights is exact, but the continuous relaxation is only qualitatively related to the prototype formulation.

The authors explicitly leave the remaining mathematical justification unresolved, specifically the formal derivation connecting the discrete prototype weights to the continuous latent-space expression. Establishing such a connection would strengthen the theoretical basis of the effective-sample-size and risk-bound analysis.

References

The cluster-to-sample bridge is exact and given by Eq.~eq:n_eff_cluster; the remaining continuous relaxation from $\alpha_k$ to $w(z)$ (e.g., via local Lipschitz arguments) is left to future work.

— Rethinking Data Augmentation under Covariate Shift: Invariant-Guided Diffusion and Prototype Reweighting  (2610.00873 - Cao et al., 1 Oct 2026) in Section 5.1, subsection “Theoretical Justification,” remark “relating $w(z)$ to $\alpha_k$”