Source
Sclocchi, Favero and Wyart (2024; PNAS 2025), LIT-884, §§3–4, Eq. 8, Figs. 5–7, SI 2–3 (Eqs. 19–59, Figs. S2–S7), as read in NOTE-685.
What was actually shown
The setting. The Random Hierarchy Model of LIT-877: v classes and v symbols per level, branching s, depth L, m rules per symbol drawn without replacement from the v^s tuples, no tuple with two parents. f, about m/v^(s−1), is the fraction of tuples a level uses. Each leaf is corrupted to belief 1 − ε in its true symbol, ε spread uniformly; with the rules known, the exact posterior is belief propagation on the tree.
The calculation. Averaging one upward BP step over draws of the rules leaves messages of the same two-valued form, so the belief p in the true symbol obeys p′ = F(p) = (p^s + f (m−1)/(mv−1)(1 − p^s)) / (p^s + f(1 − p^s)). Since F′(1) = s f m(v−1)/(mv−1), the fully informed point p = 1 is stable exactly when s f < 1 + O(1/v). Then a third fixed point between 1/v and 1 divides initial noises that flow to recovery from those that flow to chance, a transition at ε*(s f) as L → ∞. When s f > 1, every ε > 0 flows to chance. Leaves have no such flow, so their reconstruction falls smoothly in ε, and above ε* the levels’ “unchanged” curves cross: the class has changed while many leaves have not.
What could have come out otherwise. Exact BP on single draws of the rules (v = 32, s = 2, L = 10) could have departed from the annealed map; it matches the map for the class and the leaves (Figs. 6, 7a, S4, S6), and the phase diagram’s boundary agrees with ε*(s f). Gaussian diffusion of one-hot leaves, mapped to an effective ε(t), gives the same shape (Fig. S7).
What this does not say
- Not proved. The threshold is the fixed-point structure of an annealed approximation that drops the fluctuations of the random rules; its fidelity to quenched BP is numerical, for one or two parameter sets and s = 2 only.
- Not about trained diffusion models. The denoiser knows the grammar. That a diffusion model trained on images computes something like this is an analogy from the shapes of classifier-activation curves (Fig. 2); nothing here shows a trained network performs belief propagation.
- Not for every hierarchical grammar. The sparsity condition is essential: with f = 1, the maximal case of LIT-877 where every string is grammatical, the class is lost at any noise on deep trees. Results about the model in that regime, such as THEORY-195’s, and results in this one should not be pooled without stating f.
- Not that images are generated by such a grammar. The ImageNet observations are consistent with it; the Gaussian-mixture contrast rules out one alternative only.
- Not about hierarchies of attributes. The hierarchy is part–whole, as in THEORY-195; it leaves QUESTION-025 where it was.
- Not an instruction. It describes what an ideal denoiser can recover, not how to build or train one.
Connections
It is an inference-side companion to THEORY-195 on the same grammar (that one counts the examples needed to learn the grammar; this one assumes it learned). Unlike THEORY-186, where hierarchical levels separate in training time from coarse to fine, here they separate in noise, and the coarsest level is the one that changes abruptly. The threshold is a problem of reconstruction on trees, of the kind Mossel studied; the criterion here is the stability of the informed fixed point, not a proved reconstruction bound.