Pankaj Mehta and David J. Schwab (2014), arXiv preprint — ARXIV-1410.3831
Key takeaways
- The mapping is an identification of two parametrizations. Kadanoff’s variational RG couples hidden spins h to physical spins v through a kernel T_λ(v, h) and defines the coarse Hamiltonian by e^{−H^RG(h)} = Tr_v e^{T(v,h) − H(v)}. An RBM has an energy E(v, h). Setting T = −E + H (Eq. 18) makes H^RG equal the Hamiltonian of the RBM’s hidden marginal (Eq. 21), for every parameter value and, as the authors note, for any Boltzmann machine. Nothing is trained in that statement; it holds by construction.
- Exactness coincides, approximation does not. With the same identification, Kadanoff’s exactness condition Tr_h e^T = 1 for every v is the statement that the RBM’s visible marginal equals the data distribution, KL divergence zero (Eq. 22). Short of that point the paper says the two “employ distinct variational approximation schemes”: RG works on free energies, the RBM on the KL divergence. So the mapping does not say that minimizing KL performs RG.
- The 1D example is built, not learned. The decimation recursion tanh J^(n+1) = tanh² J^(n) is written as a deep network whose layers are the spins kept at each decimation step (Fig. 2). It shows that an RG scheme can be laid out as a layered network; no training objective appears.
- The 2D example is qualitative. A four-layer stacked RBM (1600, 400, 100, 25 units) trained by contrastive divergence with an L1 penalty on 40 × 40 Ising samples at J = 0.408, just above T_c, has hidden units whose effective receptive fields are local blocks that grow with depth, and reconstructs samples from 25 top units. No coarse Hamiltonian, flow, fixed point or exponent is extracted, and the L1 strength was chosen to rule out all-to-all couplings.
Standing in the record
Filed on 2026-10-09 at the owner’s request, as one of the works cited by the batch on hierarchy and hyperbolic geometry (nucleation#113, LIT-865 to LIT-877) that neither record held. It is cited by Koch-Janusz and Ringel (LIT-873), and it is filed as the position LIT-873 disputes: that paper’s supplement says its theory and its dimer counterexample “disprove the very general claims” of this one, because an RBM trained to fit the data distribution latches on to decoupled noise and misses the RG-relevant variables. Read on its own merits (NOTE-683), with the dispute assessed there on both texts. In short: the identity of Eqs. 18–22 is correct and LIT-873 does not engage it; what LIT-873 refutes is the reading of that identity as evidence that unsupervised training performs RG, a reading this paper’s title and abstract invite and its own Section III half-withdraws. The reading produces THEORY-197, the counterpart of THEORY-194 from LIT-873.
It does not bear on QUESTION-025: its hierarchy is the nesting of spatial blocks in a deep network, not attributes that imply one another.
The other side of the triangle LIT-873 draws, Lin, Tegmark and Rolnick’s Why does deep and cheap learning work so well? (arXiv:1608.08225), which denies the link, is not held in either record. McKenzie’s survey of emergence (LIT-141, NOTE-124) names this paper among its pointers.