Effective-rank control for covariance plug-in certification

Prove an effective-rank bound controlling the error of the empirical covariance estimator along the test-dependent direction used in the per-prediction nearest-centroid certificate, thereby making the plug-in variance quantities statistically self-contained.

Background

The per-prediction agreement result uses population variances and the population direction from a test embedding to a class mean. In the experiments these quantities are replaced by empirical covariance estimates and empirical directions, but the paper does not prove the concentration result required to justify that substitution in a self-contained manner. Establishing an effective-rank bound would connect the theoretical per-prediction certificate to the plug-in procedure actually evaluated.

References

A self-contained statement needs an effective-rank bound on ${\widehat\Sigma_c-\Sigma_c}_{\mathrm{op}}$ along that direction, which we do not prove here; the Hilbert-space covariance estimators and confidence balls of \citet{PaperIII} are the natural source for it.

— COMPLEX: A Closed-Form Certified Embedding of Multiparameter Persistence Modules  (2609.22012 - Majhi et al., 18 Sep 2026) in Appendix, Section “The per-prediction certificate”