Characterize the recurrent rate–distortion frontier under nonzero distortion

Characterize the full recurrent rate–distortion frontier for deterministic recurrent realizations under nonzero behavioral distortion, beyond the instantaneous rate–distortion lower bound, while accounting for the coupling of encoders across time.

Background

The paper characterizes zero-distortion recurrent behavioral memory through compatibility relations and closed compatible state assignments. For positive distortion, it establishes only that the recurrent rate is non-increasing in the distortion level and is bounded below by the instantaneous conditional rate–distortion function.

The recurrent problem is harder because the encoder must be a deterministic sequential realization whose states and updates are coupled across time. The paper explicitly leaves open whether a complete recurrent rate–distortion characterization, including properties such as convexity or equality with the instantaneous frontier, can be obtained.

References

For nonzero distortion, the instantaneous rate--distortion function remains a lower bound; we do not characterize the full recurrent frontier (Appendix~\ref{app:rdprop}).

— Minimal Recurrent Behavioral Memory for Imitation under Partial Observability  (2609.25757 - Li et al., 22 Sep 2026) in Section 2.3, “Why the general case is harder”; Appendix A.5, Section “Nonzero distortion”