Contribution
A two-page reply by the authors of LIT-882 to the appendix of Lin and Tegmark’s arXiv v1 (LIT-887). It settles what the mapping claimed about exactness: exact variational RG means the pointwise trace condition, not a matched free energy, and the authors withdraw the converse direction of their Eq. 8. It shows that Lin and Tegmark’s counterexample fails the trace condition, and so does not touch the mapping’s exact step. It adds no new result; its value is that it fixes, in the authors’ own words, the scope NOTE-683 and THEORY-197 later reached by reading LIT-882.
Key insight
The trace condition Tr_h e^{T(v,h)} = 1 for every v and the condition ΔF = 0 are different strengths of the same requirement: the first is pointwise and preserves the whole visible distribution, the second is a single number. The first implies the second, not conversely. Lin and Tegmark’s counterexample lives in the gap, and the mapping does not.
Assumptions
- Kadanoff’s variational RG with a coupling operator T(v, h) between visible and hidden variables; the coarse Hamiltonian is defined by e^{−H^RG(h)} = Tr_v e^{T(v,h) − H(v)}, as in LIT-882 Eq. 6.
- The identification T = −E + H of LIT-882, applied by the comment to Lin and Tegmark’s joint Hamiltonian, read as E(y, y′).
- Exact RG is taken to mean the trace condition (Eq. 1 of the comment, Eq. 9 of LIT-882), not ΔF = 0.
Key results
- Agreement. “A coarse-graining transformation that preserves the free energy does not suffice to reconstruct the empirical (microscopic) probability distribution … We entirely agree with this statement and never claimed otherwise.” Supported by a quoted passage from Kadanoff, Houghton and Yalabik (1976) that the variational principles pertain to the free energy and do not guarantee its derivatives.
- The trace condition implies a preserved free energy. Stated without proof; it is one line: Tr_h Tr_v e^{T−H} = Tr_v e^{−H} Tr_h e^T = Z. And it leaves the visible distribution unchanged: the joint e^{T−H}/Z marginalizes over h to e^{−H}/Z. Both checked.
- Eq. 8 withdrawn in one direction. “It is possible to preserve the free energy while violating the trace condition (note the typo in eq. (8) of [1] – it is not a biconditional. This typo possibly contributed to this misunderstanding).”
- The counterexample fails the trace condition (Appendix). With H(y, y′) = H(y) + H(y′) + K(y) + ln Z̃ and T = −H(y, y′) + H(y), −T(y, y′) = H(y′) + K(y) + ln Z̃, so Tr_{y′} e^T = Z e^{−K(y)}/Z̃, which is not constant when K is not. The comment says only that it is “non-constant”; the closed form is mine. Checked.
Claims
| id | claim | strength | support |
|---|---|---|---|
| C1 | Preserving the free energy does not suffice to recover the distribution | strong (agreed by both sides; Lin and Tegmark’s construction) | LIT-887 v1 Appendix A, accepted here |
| C2 | The trace condition implies ΔF = 0, but ΔF = 0 does not imply it; LIT-882’s ”⟺” is wrong | strong | one-line derivation; the counterexample |
| C3 | Lin and Tegmark’s counterexample violates the trace condition, so it does not refute the mapping’s Eq. 22 | strong | Appendix, checked |
| C4 | The misreading was Lin and Tegmark’s, a “misunderstanding of the variational RG procedure” | weak | LIT-882’s own text infers the trace condition from ΔF = 0 (see below); the misreading followed the paper |
| C5 | Through the trace condition, variational RG preserves more of the distribution than the free energy, as any reasonable RG must | moderate as stated; it preserves the visible marginal, not a choice of coarse variables | asserted; see Assessment |
Concepts
- trace condition: Tr_h e^{T(v,h)} = 1 for every visible configuration v; Kadanoff’s exactness condition, and LIT-882’s Eq. 9.
- preserving the free energy: ΔF = F^h − F^v = 0, i.e. the coarse partition function equals the original one.
Connections
It answers Lin and Tegmark’s arXiv v1 (LIT-887, read in NOTE-688), and is in turn answered by their v2 of 28 September 2016, which records the reply, keeps the counterexample, and adds that “any two systems that do not interact with each other will trivially satisfy their trace condition”. The v3 and journal versions of LIT-887 drop the appendix and with it the exchange. It defends LIT-882 (NOTE-683) on its own terms.
Assessment
- Against Lin and Tegmark’s v1, the reply is right. The counterexample refutes the ”⟺” of Eq. 8 and nothing else; Eq. 22 is derived from the pointwise trace condition, which the counterexample fails. NOTE-688 reaches the same verdict.
- The “typo” carried weight in LIT-882. LIT-882 introduces the exactness condition by inference: “∆F = 0 ⟺ Tr e^T = 1 (8). Thus, for any exact RG transformation, we know that Tr e^T = 1 (9)”. “Thus” runs through ΔF = 0 ⇒ trace condition, the direction the comment withdraws. Read as LIT-882 wrote it, an exact RG step was one with ΔF = 0, and Lin and Tegmark’s counterexample was aimed at exactly that reading. So C4 overreaches: the misunderstanding the comment blames on Lin and Tegmark followed the paper’s text. The repair is simple and costs nothing downstream: define exactness as Eq. 9, and Eqs. 18–22 stand.
- What the reply concedes about the approximate regime. Its own quotation from Kadanoff says the variational principle pertains to the free energy. So short of exactness, the criterion variational RG optimizes is ΔF, and the trace condition is what is approximated. Under the identification T = −E + H, NOTE-683 found that ΔF reduces to log Z − log Z_λ, a normalization only. The comment does not take this up, and by placing all of the mapping’s force on the trace condition it leaves the mapping with content only at the exact point, which is what THEORY-197 says.
- What the reply does not answer. C5 says the trace condition preserves “more information about the distribution”, and it does: the whole visible marginal. But the question Lin and Tegmark raised next is what makes h a coarse-graining of v, and the trace condition is silent on that: hidden variables independent of v satisfy it (THEORY-202). The comment itself frames RG as aiming “to preserve the long wavelength part of a distribution”, a stated target, which is the thesis of LIT-887 and THEORY-194; it does not say how the trace condition or an RBM’s training supplies that target.
- Small errors. The Northwestern affiliation carries the postcode 08854, which is New Jersey’s, not Evanston’s. No bearing on the content.
Bearing on the record
- The record’s description of the concession checks out. NOTE-683 (v2 history) says the authors “conceded Eq. 8’s ‘iff’ as a typo in arXiv 1609.03541”, and THEORY-197 (v2) that “Mehta and Schwab conceded the ’⟺’ as a typo”. Both are accurate: the comment says Eq. 8 “is not a biconditional” and calls that a typo. Two refinements, neither an error: the comment does not concede NOTE-683’s further point that under the identification ΔF measures only normalization; and its “typo” label understates, since LIT-882 inferred its exactness condition through the withdrawn direction. The comment is by Schwab and Mehta, in that order.
- THEORY-197: supported. The mapping’s own authors locate its content at the trace condition, and their quotation from Kadanoff puts the approximate criterion at the free energy, the criterion NOTE-683 shows degenerates under the mapping. It could join THEORY-197’s sources as the authors’ statement of scope.
- THEORY-202: the comment is the position its second half answers. It shows why the first half (a matched partition function is not enough) was accepted by both sides, and it asserts, without argument, that the trace condition is enough for “any reasonable RG procedure”, which THEORY-202 denies. Nothing here weakens THEORY-202.
- THEORY-194: consistent; “preserve the long wavelength part” is a stated relevance target, and the comment gives no unsupervised source for it.
- QUESTION-025: no bearing. There are no attributes, co-occurrence statistics or lattices.
- No THEORY. The reading yields no claim the record does not already hold in THEORY-197 and THEORY-202.
- Anthology. Theory of what Boltzmann machines compute, with no
practice instruction; the
analysis-and-evaluationtopic could hold it, as for LIT-882.
Limitations
- A two-page comment: no new derivation beyond the appendix check, and the free-energy implication is stated, not shown.
- It addresses only the v1 appendix and explicitly declines Lin and Tegmark’s other claims about RG and neural networks.
- It does not engage what the variational criterion becomes under the RBM parametrization, or whether the trace condition selects a coarse-graining.
- Never published outside arXiv; no later version.
Open questions
- Is there a condition, stronger than the trace condition and weaker than a stated target, that makes the hidden variables of an exact step a coarse-graining rather than an unrelated system? THEORY-202 says the trace condition is not it; the comment assumes it is.
- Did the authors ever correct Eq. 8 in LIT-882 itself? arXiv shows no v2 of LIT-882, so as of this reading the correction lives only here.