General I–MMSE relation for planted subgraph models

Establish an I–MMSE relation for the planted subgraph model that applies to arbitrary graph sequences H=H_n and characterizes the connection between MMSE discontinuities and first-order discontinuities of the model’s free energy.

Background

The paper discusses the weak-recovery or condensation threshold of the planted subgraph model and its relationship to discontinuities in the MMSE curve. In statistical physics, an I–MMSE relation often connects such MMSE behavior to discontinuities of the free energy.

For the general planted subgraph model considered here, only a one-sided I–MMSE relation is cited as known. The authors explicitly state that a general relation covering all graph sequences H is not currently known, leaving open the task of establishing such a two-sided or full correspondence.

References

Interestingly, no such I-MMSE relation is currently known for the planted subgraph model for all $H$, despite multiple partial results for specific choices of $H$.

— The Fundamental Limits of Recovering Planted Subgraphs  (2503.15723 - Lee et al., 19 Mar 2025) in Section 1, Introduction, paragraph beginning “Notably, the weak recovery threshold”