Analyze the Jacobian dynamics of MEQ for more than two modalities

Analyze the behavior and theoretical properties of the full block Jacobian for the complete-graph generalization of Mutual EQuilibrium (MEQ) to more than two interacting modalities.

Background

For more than two modalities, the paper proposes a complete-graph composition in which each modality-specific encoding function receives all modality states. In this setting, the joint Jacobian contains N-squared block Jacobians, where N is the number of modalities.

The two-modality Jacobian analysis used in the main paper no longer decomposes into the interpretable form given by the paper's lemma. Consequently, the theoretical behavior of the larger coupled system is left unresolved and is explicitly deferred to future work.

References

Unfortunately, the standard Jacobian analysis we used in our work no longer yields simple solutions under this scheme. The Jacobian $J_T*$ now consists of $N2$ block Jacobians, which means $(I-J_T*){-1}$ no longer decomposes into nicely interpretable form given by Lemma~\ref{eq:lem1}. Hence, future works should further analyze the behavior of this larger Jacobian.

— Mutual Equilibrium: Multimodal Representation Learning through Reciprocal Feedback  (2609.39456 - Park et al., 30 Sep 2026) in Appendix, Section 2, “More than Two Modalities,” paragraph “Complete graph composition”