Convergence analysis of the alternating refinement algorithm

Establish a full theoretical analysis of the alternating refinement in Algorithm~\ref{alg:Algorithm}, including a one-step contraction bound for its iterative map and accounting for the propagation of membership-update perturbations caused by continuous mixed-membership estimation.

Background

The paper develops nonasymptotic guarantees for a two-step estimator, but the practical estimator is an alternating algorithm that repeatedly updates low-rank basis matrices and continuous membership vectors. The authors explain that extending the theory from the two-step estimator to this iterative refinement is substantially more difficult than the corresponding analysis for discrete low-rank clustering, because small errors in the current basis estimates can alter the updated memberships and propagate through subsequent iterations.

The unresolved problem is therefore to prove a contraction or comparable stability result for the full alternating procedure, which would establish formal convergence and statistical guarantees for the algorithm actually used in practice rather than only for the two-step estimator.

References

Our current theory is centered on the two-step estimator, which already captures the main statistical difficulty of the problem. Extending the analysis to the alternating refinement in Algorithm~\ref{alg:Algorithm} appears to be substantially more delicate, as it would require a one-step contraction bound for the iterative map. This is harder in the mixed-membership setting than in discrete clustering in since the membership update is continuous rather than a hard assignment, and small perturbations in the current basis estimates can still change the updated memberships and hence propagate across iterations. Another related piece is the algorithm in , which is more amenable to a direct perturbation analysis. However, its guarantee is based on HOOI followed by simplex recovery rather than on controlling the error in an iterative refinement with simplex constraint. A full analysis of our current Algorithm~\ref{alg:Algorithm} is therefore left for future work.

— Mixed Membership Model of Low-rank Matrices with Multimodal Extension  (2608.18953 - Snider et al., 19 Aug 2026) in Discussion section