Source
Mehta and Schwab (2014), LIT-882: Sections I and III, Eqs. 1–22, and the closing paragraph of Section III; as read in NOTE-683. Koch-Janusz and Ringel (2017; Nature Physics 2018), LIT-873: main text p. 4 and the supplement’s comparison with contrastive-divergence RBMs, as read in NOTE-674.
What was actually shown
The identity. Kadanoff’s variational RG couples hidden spins h to physical spins v by a kernel T(v, h) and defines the coarse Hamiltonian by e^{−H^RG(h)} = Tr_v e^{T(v,h) − H(v)}. An RBM has a joint energy E(v, h). With T = −E + H, H^RG equals the Hamiltonian of the RBM’s hidden marginal, for every parameter value and for any Boltzmann machine (Eq. 21). And e^T = p(h|v) e^{H − H^RBM}, so the exactness condition Tr_h e^T = 1 for every v holds exactly when the RBM’s visible marginal is the data distribution (Eq. 22). Both are short, checked derivations.
The concession. The paper states that short of exactness RG works on free energies and the RBM on the KL divergence, and that the two “employ distinct variational approximation schemes”.
The free energy under the identity (the reader’s derivation from Eqs. 4, 6, 7 and 18, in NOTE-683): F^h = −log Z_λ, so ΔF = log Z − log Z_λ, which a constant shift of E sets to zero for any parameters. The criterion variational RG minimizes therefore measures only normalization once the kernel is the RBM’s, and the paper’s Eq. 8, which equates ΔF = 0 with the pointwise exactness condition, is too strong: ΔF = 0 says only that the condition holds on average under the data. Mehta and Schwab conceded the ”⟺” as a typo in their comment on Lin and Tegmark (arXiv 1609.03541), and Lin and Tegmark’s v2 appendix makes the normalization point in substance (LIT-887, NOTE-688).
The counterexample. LIT-873 trains an RBM by contrastive divergence on fully packed dimers with added decoupled spin pairs; it spends its hidden units on the noise, then on columnar textures, not on the staggered patterns that carry the RG-relevant electric fields.
What could have come out otherwise: the identity could have required conditions on E or on training; it does not. The counterexample’s RBM could have found the fields.
What this does not say
- Not that the correspondence is wrong. Eqs. 18–22 hold, and nothing in LIT-873 disputes them.
- Not that distribution-fitting networks never perform RG. One counterexample shows they need not; when they do is open. The stacked, L1-penalized network of LIT-882 was not tested on LIT-873’s noisy dimers.
- Not that variational RG in Kadanoff’s own practice has no approximate criterion. The claim is about the criterion as the mapping paper states it (minimize ΔF), under the mapping’s kernel. Kadanoff’s lower-bound schemes, as usually described, keep the trace condition exactly and approximate elsewhere; how they fare under the RBM parametrization is not worked out in either paper.
- Not a claim about deep networks other than Boltzmann machines, nor an instruction for machine learning.
- Nothing about attribute hierarchies or embeddings. The levels are nested spatial blocks.
Connections
It is the formal counterpart of THEORY-194, the empirical account from LIT-873 that the coarse-graining informative about the far environment is the RG-relevant one while a distribution-fitting one is not. Together they say: the formalisms correspond exactly, and which variables are kept is fixed by the objective, not by the correspondence.
Note of 2026-10-19: the correspondence, re-derived
Write p(v) = e^{−H(v)}/Z for the data. The RBM has joint energy E(v, h) and marginal energies e^{−H^RBM(h)} = Tr_v e^{−E} and e^{−H^RBM(v)} = Tr_h e^{−E}, with partition function Z_λ.
- The identity. With T = −E + H, Tr_v e^{T − H} = Tr_v e^{−E} = e^{−H^RBM(h)}. So H^RG = H^RBM for every parameter value. Nothing about training or the form of E is used.
- Exactness. Tr_h e^T = e^{H(v) − H^RBM(v)}, which is 1 for every v iff H^RBM = H. Then Z_λ = Z and p_λ = p, so an exact step is a perfect fit. The converse holds only up to a constant. A perfect fit gives H^RBM = H + log(Z_λ/Z), and the trace condition holds once E is shifted by that constant, which leaves p_λ unchanged.
- Short of exactness. F^h = −log Tr_{h,v} e^{−E} = −log Z_λ, so ΔF = log Z − log Z_λ. A constant shift of E sets ΔF to zero for any parameters. Under the mapping’s kernel, the criterion variational RG minimizes measures only normalization, while the RBM minimizes the KL divergence. ΔF = 0 is the trace condition only on average under the data, which is why Eq. 8’s converse fails.
- The negative half. It is a non-implication, so one counterexample suffices. LIT-873’s contrastive-divergence RBM fits the distribution of dimers with decoupled spin pairs and keeps the wrong variables.