Exact recovery guarantee for SC-LCM under strengthened conditions

Prove that the spectral clustering algorithm SC-LCM achieves exact recovery of the latent class membership matrix Z (up to label permutation) in the latent class model for ordinal categorical data (LCM(K) in Definition 1) under appropriately strengthened signal and separation conditions, by developing the required detailed entrywise singular subspace perturbation analysis so that Assumption A5 holds for SC-LCM.

Background

Assumption A5 in the paper requires an estimator that achieves exact recovery of the class membership matrix when the candidate number of classes equals the truth. The paper adopts SC-LCM as a practical estimator and proves only its consistency (vanishing misclassification rate), not exact recovery.

Recent theoretical advances (e.g., leave-one-out singular subspace perturbation analysis and exact recovery results for the binary SOLA algorithm) suggest that exact recovery for SC-LCM might be achievable under stronger conditions. However, carrying out a rigorous, entrywise analysis for SC-LCM in the ordinal setting is beyond the scope of this paper, and the authors explicitly defer this question to future work.

References

This indicates that SC-LCM could potentially be shown to achieve exact recovery under analogous strengthened conditions. A rigorous proof of this, however, would require a detailed entrywise analysis that is beyond the scope of the present paper, whose main focus is to develop a goodness-of-fit test for the number of latent classes. We leave this meaningful theoretical question for our future work.

— Goodness-of-Fit Tests for Latent Class Models with Ordinal Categorical Data  (2602.21572 - Qing, 25 Feb 2026) in Section 3.2 (Practical test statistic), discussion around Assumption A5