NOTE-683: An exact mapping between the Variational Renormalization Group and Deep Learning

Contribution

It writes Kadanoff’s variational real-space RG and the restricted Boltzmann machine in one notation and shows that, under the identification T = −E + H of the RG kernel with the RBM energy, the coarse-grained Hamiltonian of RG and the marginal Hamiltonian of the RBM’s hidden layer are the same function, and that Kadanoff’s exactness condition is the same as the RBM reproducing the data distribution exactly. It adds a deep network that carries out decimation of the 1D Ising chain by construction, and an observation that a stacked RBM trained on near-critical 2D Ising samples has local receptive fields that grow with depth. It was, as far as its references show, the first statement of the RG–deep-learning correspondence in this form, and the paper later work on the question answers to.

Key insight

A coarse-graining that integrates out visible spins against a coupling kernel and an RBM that marginalizes its visible layer are the same operation written twice: once the kernel is defined as the RBM’s energy plus the data Hamiltonian, the two coarse Hamiltonians cannot differ. What the identity leaves open is the thing RG is for: which kernel to choose. RG chooses it to preserve the long-distance physics; an RBM chooses it to reproduce the data. The two choices coincide only where both are exact.

Assumptions

• Binary spins, Boltzmann data. P(v) = e^{−H(v)}/Z with temperature set to one; H is a general polynomial in the spins (Eq. 3).
• RBM energy E(v, h) = Σ b_j h_j + Σ v_i w_ij h_j + Σ c_i v_i (Eq. 10; the sign convention is the paper’s), joint p_λ = e
• The identification T(v, h) = −E(v, h) + H(v) (Eq. 18) is a definition, not a result. It makes T depend on the data Hamiltonian, so T is not a fixed functional form in λ applied across a family of Hamiltonians, as a Kadanoff kernel usually is.
• Exact RG is taken to mean Tr_h e^{T(v,h)} = 1 for every v (Eq. 9).
• The 2D experiment. Periodic 40 × 40 Ising lattice at J = 0.408 (paramagnetic side); equilibrium Monte Carlo samples (20,000 in the text, 40,000 in Appendix A); four RBMs of 1600, 400, 100 and 25 units trained layer by layer by contrastive divergence for 200 epochs with an L1 penalty of 2 × 10⁻⁴, chosen “to ensure that one could not have all-to-all couplings”; no convolution.

Key results

• Eq. 21. With T = −E + H, H^RG_λ(h) = H^RBM_λ(h) for every λ: the RBM’s hidden marginal is of Boltzmann form with the RG Hamiltonian. The proof is two substitutions (Eqs. 19–20). The authors note it does not use the form of E, so it holds for any Boltzmann machine.
• Eq. 22. e^{T(v,h)} = p_λ(h|v) e^{H(v) − H^RBM_λ(v)}. Hence when Tr_h e^T = 1 for every v, the RBM’s visible Hamiltonian is the data Hamiltonian, D_KL(P‖p_λ) = 0, and T is exactly the log of the conditional p_λ(h|v). (The converse also holds, up to an additive constant in E.)
• Approximations differ (end of Section III, the paper’s own words): variational RG schemes “work at the level of the Hamiltonians and Free Energies”, whereas RBMs minimize the KL divergence, so “the two approaches employ distinct variational approximation schemes for coarse graining”.
• 1D Ising (Fig. 2). Decimating every other spin gives tanh J^(n+1) = tanh² J^(n), flowing to the stable fixed point J = 0. Written as a layered network with interlayer weights J^(n), marginalizing a layer is one decimation. The authors note it “contains no information about half of the visible spins”.
• 2D Ising (Fig. 3). Effective receptive fields (products of the weight matrices, Appendix B) of the 100 middle and 25 top units are local blocks of roughly equal size within a layer, larger in higher layers; reconstructions from the top layer keep the samples’ macroscopic domains, at a compression of 64.

Claims

id	claim	strength	support
C1	Under T = −E + H, the RG coarse-grained Hamiltonian equals the RBM hidden-marginal Hamiltonian, for any parameters and any Boltzmann machine	strong (proof), but true by construction	Eqs. 18–21
C2	An exact variational RG step (Tr_h e^T = 1) is the same as an RBM fitting the data distribution exactly	strong (proof)	Eq. 22
C3	ΔF = 0 is equivalent to the exactness condition (Eq. 8)	not supported; false in general	Eq. 8 asserts ”⟺”; see Limitations
C4	There is a “one-to-one mapping” between RBM-based deep networks and variational RG	moderate as a correspondence of model classes; not as a correspondence of procedures	C1–C2, with the paper’s own concession that the approximate schemes differ
C5	A deep network can implement decimation of the 1D Ising chain	strong, by construction	Fig. 2, Eq. 24
C6	A stacked RBM trained on near-critical 2D Ising samples self-organizes into block-spin-like coarse-graining	weak	one temperature, one lattice size, qualitative receptive fields, locality encouraged by L1; Fig. 3
C7	Deep learning may be employing a generalized RG-like scheme	weak (hedged suggestion)	C1–C6

Method

Section I reviews Kadanoff’s variational RG: a kernel T_λ(v, h), the coarse Hamiltonian from Eq. 6, the free energies F^v and F^h_λ, and the choice of λ by minimizing ΔF = F^h_λ − F^v. Section II reviews RBMs, their marginals and KL training, and stacking. Section III defines T by Eq. 18 and derives Eqs. 19–22. Section IV gives the 1D construction analytically and the 2D experiment numerically, using Hinton and Salakhutdinov’s stacked-RBM code (unsupervised phase only).

Concepts

• variational RG: Kadanoff’s real-space scheme in which hidden spins are coupled to physical spins through a parametrized kernel T_λ, and λ is chosen variationally; here, by minimizing ΔF.
• exact RG transformation: one with Tr_h e^{T(v,h)} = 1 for every configuration v, so that the partition function is preserved.
• H^RG_λ, H^RBM_λ: the coarse Hamiltonian defined by tracing out v against T, and the Hamiltonian of the RBM’s hidden marginal p_λ(h).
• effective receptive field: r^(l) = r^(l−1) W^(l) with r^(1) = W^(1), the product of the weight matrices up to layer l; a measure of which visible spins a hidden unit is coupled to through the stack.

Connections

It rests on Kadanoff’s variational RG (Kadanoff, Houghton and Yalabik 1976; Efrati, Wang, Kolan and Kadanoff 2014, which reads tensor-network methods as variational RG) and on Hinton’s stacked RBMs and contrastive divergence. Within this record its sequel in argument is Koch-Janusz and Ringel (LIT-873), and the account produced from that reading is THEORY-194. The third position, Lin, Tegmark and Rolnick’s denial that RG and unsupervised learning are related (arXiv:1608.08225), is not held.

The dispute with LIT-873, on both texts

What LIT-873 says. Main text p. 4: its dimer filters “are orthogonal to those obtained using Kullback-Leibler (KL) divergence”, which “shows that standard RBMs minimizing the KL-divergence do not generally perform RG, thereby contradicting prior claims” (this paper). The supplement (pp. 17–18): this paper “claims an ‘exact’ mapping”; “our theoretical arguments, as well as explicit numerical results on dimer model disprove the very general claims” of it; “a neural network is not specified without the cost function”, and this paper assumes the KL cost explicitly. The evidence is one CD-trained RBM on fully packed dimers with added decoupled spin pairs: with few hidden units it couples to the noise pairs, with many it learns columnar rather than staggered dimer textures.

What this paper claims, read exactly. The proved content is C1 and C2. The title says “exact mapping” and the discussion “one-to-one mapping”, but the interpretive claims are hedged: the mapping “suggests” that deep networks implement “a generalized RG-like procedure”, and the 2D result “suggests the DNN is self-organizing to implement block spin renormalization”. The paper does not state that every KL-trained RBM performs RG, and at the end of Section III it says in terms that, short of exactness, RG and RBM training “employ distinct variational approximation schemes”.

Assessment.

• On the identity, this paper stands and LIT-873 does not touch it. Eqs. 18–22 are correct, and LIT-873 neither disputes nor engages them. Its word “disprove” applies to the interpretation, not to the mathematics.
• On what the identity shows, LIT-873 is right, and for a reason this paper’s own text supplies. The identity holds for every λ, trained or not, good RG or bad, so it cannot by itself select a coarse-graining. The only point at which RG and RBM training provably coincide is the exact one, and an RBM with a restricted number of hidden units cannot match an arbitrary distribution exactly, as the paper notes citing Le Roux and Bengio (the bound it quotes is 2^N hidden units; a coarse-graining has fewer than N). Everywhere training actually operates, the two procedures optimize different things, which is LIT-873’s “the cost function decides” and this paper’s “distinct variational approximation schemes”. LIT-873 adds what the concession lacked: a case where the difference changes which variables are kept, and an objective (mutual information with the environment beyond a buffer) that keeps the right ones.
• The point is sharper than either paper says. By my derivation from Eqs. 4, 6, 7 and 18 (reached independently; Lin, Tegmark and Rolnick’s v2 appendix, LIT-887, makes the same point in substance): under the identification, F^h_λ = −log Z_λ, so ΔF = log Z − log Z_λ. Variational RG’s own criterion, ΔF, then measures only the RBM’s normalization; it vanishes for any λ once E is shifted by a constant, and carries no information about how well the RBM fits or coarse-grains. In general ΔF = 0 says only that ⟨Tr_h e^T⟩_P = 1, an average, so Eq. 8’s ”⟺” with the pointwise condition is wrong. The mapped variational RG therefore has no non-trivial approximate criterion of its own; the only criterion left on the RBM side is KL, which is the one LIT-873 shows can keep the wrong variables.
• On the evidence, LIT-873’s counterexample does not refute this paper’s 2D observation, and does not need to. The setups differ: one CD-trained RBM on noisy dimers there, an L1-penalized four-layer stack on clean Ising here. Clean Ising near T_c has no strongly patterned short-range noise to latch on to, and block-like receptive fields there are what local correlations and an L1 penalty set to forbid all-to-all couplings would give in any case; growing receptive fields follow from composing sparse local weight matrices. So this paper’s 2D result was weak evidence for RG before LIT-873 and is not contradicted by it; LIT-873 shows the inference from it to “deep learning does RG” does not generalize.
• Where LIT-873 overreaches. “Contradicting prior claims” and “very general claims” read this paper at its title and abstract rather than at its hedged conclusions and its own concession. Its counterexample is one dataset, one architecture and one training algorithm; it shows that distribution fitting need not give RG, not when it does (NOTE-674’s C4 and limitations already say so).
• Verdict. On the question actually in dispute, whether unsupervised training by distribution fitting performs RG, LIT-873 is right and this paper gave no reason to think otherwise beyond a suggestive picture. On the exact correspondence of the two formalisms, this paper is right, and the correspondence is narrower than its title: exact as a re-parametrization and at the exact point, silent everywhere else.

Bearing on the record

• Produces THEORY-197: the correspondence is an identity of parametrizations whose only non-trivial content is the coincidence of the exact points; it does not make distribution-fitting training an RG step. It is the formal counterpart of THEORY-194, which holds the empirical side from LIT-873.
• THEORY-194. This reading supports it on the point it shares with this paper (which objective decides) and adds the formal reason the fitted coarse-graining need not be the RG one: the identity is silent on which kernel to choose. It does not bear on THEORY-194’s mutual information criterion.
• THEORY-036 and LIT-219, LIT-150. No bearing beyond what NOTE-674 says: this paper offers no criterion for which compression counts.
• QUESTION-025 and THEORY-185. No bearing. The hierarchy is of spatial blocks; there are no attributes, co-occurrence statistics or lattices.
• Anthology. A theory of what deep networks learn, with no practice instruction; the anthology’s analysis-and-evaluation topic, which takes theory, could hold it, hence anthology-candidate on the LIT.

Limitations

• The mapping is definitional. T is defined so that Eq. 21 holds; the identity holds for untrained and badly trained networks alike, and T depends on the data Hamiltonian rather than being a fixed kernel form.
• Eq. 8 overstates. ΔF = 0 is equivalent to ⟨Tr_h e^T⟩_P = 1, not to Tr_h e^T = 1 for every v; and under Eq. 18 it reduces to Z_λ = Z (my derivation, above).
• The 1D network is constructed from the known decimation, not learned, and the paper does not write out its T or E to show it is an instance of Eqs. 18–22.
• The 2D evidence is qualitative and single-condition. One lattice size, one temperature on the paramagnetic side, no coarse couplings, no flow, no fixed point, no exponent; locality is encouraged by the L1 penalty; no comparison with a known block-spin transformation beyond appearance.
• Small errors. The 2D critical coupling is given as J/k_BT = 0.4352; the exact value is ln(1 + √2)/2 ≈ 0.4407 (the experiment’s 0.408 is on the paramagnetic side either way). The sample count is 20,000 in the text and 40,000 in Appendix A. The SI and Matlab file cited in the appendix are not on arXiv.
• Never published in a journal; arXiv v1 is the only version.

Open questions

• When does minimizing KL with fewer hidden than visible units give an RG transformation, i.e. a coarse Hamiltonian that stays short-ranged and keeps the relevant operators? A characterization for a class of Hamiltonians would settle the scope of C4 and of LIT-873’s counterexample together.
• Does the stacked RBM of Fig. 3, run on the noisy dimer data of LIT-873, also spend its units on the noise? That is the direct test of whether depth and the L1 penalty change LIT-873’s single-RBM result.
• Is there a non-trivial approximate criterion on the RG side, short of exactness, that the RBM parametrization preserves? Under Eq. 18 the free-energy criterion does not supply one.