Source
Raju, Machta and Sethna (2018), Phys. Rev. E 98:052112, LIT-1254, arXiv:1710.05787v2, read in full in NOTE-972. The identity is the paper’s Eqs. 10, 14 and 15. The closure step is Eqs. 11–12 and the line after Eq. 15 (“g̃µν = gµν − ∂µ∂νG”). The diffusion result is Eqs. 5–9.
What was actually shown
The metric throughout is the Fisher information of the whole data set, not a density. It is extensive, so a coarse-grained system of the same physical size, with fewer coarse sites, carries less of it by construction. That accounting is the paper’s choice (its modified Lie derivative L̄ = L_β − L∂_L), and the statements below use it.
Re-derivation, step by step.
- The bare metric. Take P(x | θ) = exp(−Σ_μ θ_μ Φ_μ(x)) / Z(θ). The score is ∂_μ log P = −Φ_μ + ⟨Φ_μ⟩, so g_μν = Cov(Φ_μ, Φ_ν) = ∂_μ∂_ν log Z (Eq. 10).
- The coarse metric. For a map x̃ = C(x), p(x̃ | θ) = Σ_{x: C(x) = x̃} P(x | θ). Differentiating the sum gives ∂_μ log p(x̃) = −E[Φ_μ | x̃] + ⟨Φ_μ⟩, so g̃_μν = Cov(E[Φ_μ | x̃], E[Φ_ν | x̃]) (Eq. 15).
- The loss. By the law of total covariance, Cov(Φ_μ, Φ_ν) = E[Cov(Φ_μ, Φ_ν | x̃)] + Cov(E[Φ_μ | x̃], E[Φ_ν | x̃]). So g − g̃ = E[Cov(Φ | x̃)] (Eq. 14), which is positive semidefinite. Nothing about renormalization is used up to here.
- Closure. Suppose the restricted partition function is Z̃(x̃) = exp(G(θ) − H̃(x̃; θ̃)), with H̃ = Σ θ̃_μ Φ̃_μ(x̃) of the same form on the coarse lattice and θ̃ = Mθ linear. Then −log p(x̃) = H̃(x̃; Mθ) − G(θ) + log Z(θ). H̃ is linear in θ, so its second derivatives vanish, and taking expectations gives g̃ = g − ∂∂G. With step 3, ∂∂G = E[Cov(Φ | x̃)] ⪰ 0. (The sign of G here is chosen so that the loss is +∂∂G. The paper’s Eq. 11 writes the same relation less clearly.)
- The singular part. If G is analytic in θ, then g − g̃ is analytic, so g̃ and g have the same singular part as functions of θ. Equivalently, summing step 4 over x̃ gives log Z(θ, L) = G(θ) + log Z(Mθ, L/b), and differentiating twice gives g^(s)(θ, L) = Mᵀ g^(s)(Mθ, L/b) M. That is the finite form of the paper’s Eq. 17, L̄_β g^(s) = 0, which the paper obtains from the scaling relation Eq. 16 by differentiating twice. The term in ∂∂β drops only because β is linear.
- Diffusion. Take increments with kernel K(x) = φ(y)[1 + Σ_{μ≥3} θ_μ He_μ(y) / (μ! θ₂^{μ/2})] / √θ₂, with y = (x − θ₁)/√θ₂ and He_μ the probabilists’ Hermite polynomials. At θ_μ = 0 (μ ≥ 3), the score of one increment along θ_μ is He_μ(y)/(μ! θ₂^{μ/2}), and E[He_μ He_ν] = μ! δ_μν, so m increments carry g_μν = m δ_μν / (μ! θ₂^μ) (Eq. 6). To first order in θ the n-fold convolution multiplies the Fourier factor (ik√θ₂)^μ e^{−k²θ₂/2} by e^{−(n−1)k²θ₂/2}, which is the μ-th Hermite term of the n-step Gaussian with coefficient n·n^{−μ/2}. So the score of one n-step increment along θ_μ is n^{1−μ/2} He_μ(y⁽ⁿ⁾)/(μ! θ₂^{μ/2}), with y⁽ⁿ⁾ standard normal; its variance is n^{2−μ}/(μ! θ₂^μ); and m/n such increments give g⁽ⁿ⁾_μν = n^{1−μ} g⁽⁰⁾_μν (Eq. 9). For the drift and variance directly: an n-step increment has mean nθ₁ and variance nθ₂, so m/n of them carry (m/n)·n²/(nθ₂) = m/θ₂ about θ₁ (n⁰) and (m/n)·1/(2θ₂²) about θ₂ (n⁻¹), both matching n^{1−μ}. Only first order in θ is needed, because the metric at θ = 0 depends only on first derivatives of the distribution.
Checked numerically (a script in the reader’s scratchpad, not the record): for μ = 3, 4, 5 and n = 1, 2, 4, 8, the Fisher information per n-step increment of a numerically convolved kernel agrees with n^{2−μ}/μ! to six figures.
What could have come out otherwise. The loss could have contained a non-analytic part even for an exact step, which would have eroded the singular part; under closure and an analytic G it cannot. The diffusion scaling could have been non-monotone in μ, or could have spared the variance; it is neither.
What this does not say
- Not that relevant directions keep their information in general. The title keeps the singular part of the metric. The full metric along a relevant direction also has an analytic part, which the coarse data lose. The paper’s own scalings, Eqs. 20–21, put the singular part at L^d ξ^{−(d−2λ)} and the analytic part at L^d b^{−(d−2λ)}. For b < ξ the singular part dominates only when λ > d/2, where the susceptibility diverges. A relevant direction with 0 < λ < d/2, such as the thermal direction of a transition with α < 0, would lose information as b^{−(d−2λ)} until b reaches ξ. The paper does not state this, but Machta et al. (2013, LIT-1299) do: their supplement says the singular part dominates “provided 2y_i − d ≥ 0” and that relevant directions with 0 < 2y < d contract. This paper dropped it. The diffusion example agrees with it: the drift, kept exactly, has λ = 1/2 = d/2.
- Not that the drift is kept away from the Gaussian fixed point. The paper explains its preservation as “the mean drift is given by the two endpoints”. The endpoints are sufficient for the drift only when the kernel is Gaussian. For a unit-variance Laplace kernel, the reader’s computation gives the drift information per original step as 2.00, 1.19, 1.09 and 1.06 at n = 1, 2, 3, 4, falling towards the Gaussian bound 1/θ₂. So the non-Gaussian surplus is lost, as the paper’s general picture would expect, but the relevant direction is not “exactly preserved”.
- Not that irrelevant directions contract as b^−(d−2λ). Eq. 21 comes from a scaling argument (“should be dominated by”, “with higher order corrections”), and the authors rest their confidence on the numerics of Machta et al. (2013), filed as THEORY-777 (Proposed): one 64×64 lattice at T_c, at most six decimation steps of √2, rates read against drawn bands with no fitted exponents or error bars. It is exact only for the diffusion model above.
- Not that the closure holds. The renormalized Hamiltonian staying in the family, a linear parameter map and an analytic G are assumed. A real RG step generates new couplings, and in the exact 1D Ising decimation the map K′ = ½ ln cosh 2K is not linear.
- Not that this selects a coarse-graining. The map C is given. The identity of step 3 says how much any map loses, and that a map whose coarse state determines the energy operators loses nothing. It does not say which maps are renormalizations.
- Nothing about learning. The metric measures distinguishability from a fixed amount of data. It is not a learning rate, and no training dynamics appear in the paper.
Connections
The loss in step 3 is the coarse-graining analogue of the data-processing inequality for Fisher information, written out for exponential families. It gives THEORY-202’s “stated target” one concrete form: keep the singular Fisher metric of the bare couplings. Under that target the non-interacting construction in THEORY-202 fails, since its coarse variables carry no information about θ. The identity map passes, though, so the target is necessary, not sufficient. THEORY-194 selects coarse variables by their mutual information with the far environment, which is a different quantity from the Fisher information about the couplings measured here. Whether the coarse-graining that maximizes the former keeps the singular part of the latter has not been tested. On QUESTION-217, the contraction factors (the generalized eigenvalues of g̃ relative to g, which do not depend on the parametrization) order parameter directions from coarse to fine along a chain of nested coarse-grainings. In the diffusion case that order is by cumulant order. That is an order of relevance tied to refinement, not an order of acquisition.