Source
Koch-Janusz and Ringel (2017; Nature Physics 2018), LIT-873: main text Figs. 2, 4 and 5; supplement Eqs. 5–15 (the estimator), Eqs. 20–40 (1D Ising), Eqs. 41–44 (saturation), Figs. 8–11; as read in NOTE-674.
The sequels: Lenggenhager, Gökmen, Ringel, Huber & Koch-Janusz (2020), LIT-881, Appendix B’s theorems, as read in NOTE-682; and Gökmen, Ringel, Huber & Koch-Janusz (2021), LIT-878, the interacting dimer model, as read in NOTE-681. The formal counterpart, on why a fitted coarse-graining is not one, is THEORY-197 from Mehta & Schwab’s mapping (LIT-882).
What was actually shown
The objective. For a block V, a buffer B and the environment E beyond it, choose the coarse variables H, drawn from an RBM-form P_Λ(H|V), to maximize I_Λ(H : E). Only Monte Carlo samples are used.
Numerically, in 2D. For the Ising model the maximizer on a 2 × 2 block is Kadanoff’s block spin, and on larger blocks it weights the boundary. For the fully packed dimer model it reads the low-momentum electric fields of the height-field description, and it gives zero weight to decoupled, regularly patterned noise spins. Iterated four times on 128 × 128 Ising samples, it produces the flow away from T_c on both sides, locates T_c to about 1%, and gives ν ≈ 1.0 ± 0.15. An RBM trained by contrastive divergence to fit the noisy dimer data instead spends its hidden units on the noise, then on columnar patterns that carry no field. Each could have come out otherwise: the maximizer could have coupled to the noise, the flow could have failed to separate, or the fitted RBM could have found the fields too.
Analytically, in 1D. Among decimation, boundary-majority and uniform majority filters for the 1D Ising chain, decimation, the known optimal RG, carries twice the boundary filter’s information and the uniform filter’s vanishes with block size. For 1D and quasi-1D systems with short-range interactions, if adding hiddens no longer increases the information, the effective Hamiltonian has no coupling across the block.
What this does not say
- Not that the objective finds relevant variables in general. Every 2D case had a known answer. The account is that, on these models, the objective and RG agree; whether they agree where RG’s variables are unknown is what QUESTION-232 asks.
- Not a proof of equivalence with RG in two or more dimensions. The equivalence with a compact, short-ranged effective Hamiltonian is proved only for 1D and quasi-1D chains, and the 1D comparison is among three filters, not over all.
- Not that distribution-fitting networks never coarse-grain correctly. One dataset shows that they need not; when the two objectives coincide is open.
- Not that the variables are intrinsic to the data. The block, buffer and environment are a spatial partition supplied before anything is learned, and the buffer’s width is a choice. Mutual information is invariant under bijections of the variables, but the partition is not; so, consistently with THEORY-017, the parts the method compresses are given to it.
- Not a hierarchy of attributes. The levels are nested spatial blocks. Nothing here concerns attributes implying one another, linear attribute directions in embeddings or concept lattices, so it leaves QUESTION-025 as it was.
- Not an instruction for machine learning. It says which objective coincides with RG on these models, not how to train anything.
Connections
It gives an operational criterion for the scale-relative, lossy description that THEORY-036 and Ladyman’s real patterns (LIT-219) take RG as their example of: the compression that counts is the one that keeps information about the far field. The objective is a variant of the information bottleneck (LIT-338) with the environment as the relevance variable.
Note of 2026-10-20: scope at promotion
Promoted on one study read in full (ADR-047 v3). NOTE-674 read the figures from their captions. For the promotion they were checked against the arXiv PDF of LIT-873 (18 pp.). Fig. 2(A) shows a single hidden unit weighting a 2 × 2 block uniformly. Fig. 4 shows 8 × 8 filters with the staggered E_y and E_x ± E_y patterns. Fig. 5 shows effective temperature against RG step for starting T/T_c from 0.98 to 1.03: the curve from 0.99 flows down and the one from 1.002 flows up. Fig. 11 shows the contrastive-divergence filters coupling to the noise pairs, then to columnar textures. The 1% and ν ≈ 1.0 ± 0.15 are stated in the text, together with the method (finite-size data collapse over four RG steps, 5000 samples per temperature). The supplement’s Fig. 9 says only that the slope is “consistent with” ν = 1.
Three things are left out of the title. First, the 1D result: it compares three filters on the paper’s proxy for mutual information, to leading order in block size, and decimation scores twice the boundary filter. That does not show decimation maximizes the objective over all filters. Second, the general equivalence with renormalization, which QUESTION-232 now carries. Third, the former title’s closing gloss, that which variables a compression keeps is set by what it must stay informative about. That is the record’s reading of these results, and THEORY-202 cites this account for it. The results are consistent with it but do not test it apart from the choice of environment.
Note of 2026-10-20: the wider claim filed
The promotion audit narrowed this entry to what was shown. Under ADR-047 v4 the claim the narrowing dropped, that the mutual-information criterion finds the renormalization-group-relevant variables in general, where they are not known beforehand. The record’s gloss, that which variables a compression keeps is set by what it must stay informative about, is already carried by CLAIM-152, which grounds on this entry, is now filed as THEORY-789, Proposed, which extends this entry and takes it as evidence. This entry’s title and status are unchanged.