Mean dimension and rate-distortion function revisited

Rui Yang

REVIEW 3 major objections 5 minor 38 references

Mean dimension and rate-distortion function revisited

T0 review · 3 major / 5 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read The paper proves that for every invariant measure on the Hilbert cube shift, the lower and upper mean Rényi information dimensions coincide with the corresponding information dimension rates, answering an open question from earlier work, an

desk verdict A real answer to an open question, wrapped in a broader unification that has one load-bearing gap: the non-ergodic extension of Wang's proposition is asserted, not proved. read the letter →

arxiv 2510.08051 v2 pith:CDVSX7AK submitted 2025-10-09 math.DS cs.ITmath.IT

Rui Yang This is my paper · ORCID

classification math.DScs.ITmath.IT MSC 37A0537A3537B4094A34

keywords meandimensionrate-distortionfunctionsRényiinformationrateKolmogorov–Sinaientropyvariationalprincipleergodicmeasuresmarkerproperty

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper's central claim is that two seemingly different ways of measuring the 'information dimension' of an invariant measure on the Hilbert cube shift—one based on optimally fine partitions, the other on coordinate quantizers—always agree, even when the measure is not ergodic. This settles an open question that had previously been resolved only for ergodic measures. The paper also introduces four types of rate-distortion entropies and proves that, under reasonably general shadowing conditions, each converges to the classical Kolmogorov–Sinai entropy of the measure. Finally, for systems with the marker property and finite mean dimension, it shows that the double variational principle for mean dimension holds with the supremum taken over ergodic measures, and that this extends to a whole family of measure-theoretic ε-entropies. The upshot is a unified picture in which rate-distortion, partition entropy, and information dimension give compatible answers about the complexity of infinite-entropy systems.

What carries the argument

The load-bearing object is the mean Rényi information dimension, MRID(X,T,d,μ)=liminf_{ε→0} (inf_{diam(α)≤ε} h_μ(T,α))/log(1/ε), compared with the information dimension rate defined through coordinate quantizer partitions α_m. The proof of Theorem 1.1 bridges the two through L^2 rate-distortion functions: known results identify the information dimension rate with the L^2 rate-distortion dimension, and a new inequality (Lemma 3.1) shows rdim_{L^p} ≤ MRID, while a covering argument yields the reverse bound. For Theorem 1.2, an empirical-measure ε-entropy serves as the bridge: the proof establishes an inequality chain from that entropy to the rate-distortion entropies and then to Kolmogorov–Sin

What would settle it

Compute the L∞ rate-distortion entropy lim_{ε→0} R_{μ,L∞}(ε) for a non-ergodic measure, say μ = (1/2)δ_{0^∞} + (1/2)L^Z on [0,1]^Z, and compare it with h_μ(σ). If they differ, Theorem 1.2(2) fails for non-ergodic measures. For Theorem 1.1, compute MRID for the same μ via optimal ε-partitions and compare with d(μ); a mismatch would disprove the claimed equality.

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Extended reading notes

Core claim

The main result is Theorem 1.1: for every σ-invariant Borel probability measure μ on [0,1]^Z with the metric d_Z(x,y)=Σ_{n∈Z}|x_n−y_n|/2^{|n|}, the lower and upper mean Rényi information dimensions MRID and MRID equal the lower and upper information dimension rates d(μ) and d̄(μ). This was known for ergodic measures; the paper removes the ergodicity assumption. Theorem 1.2 introduces L^p, L^∞, Bowen, and r rate-distortion entropies and shows that for ergodic measures all equal h_μ(T), while under the g-almost product property this equality holds for every invariant measure. Theorem 1.3 states that if a system has the marker property and finite mean dimension, the double variational principle

Load-bearing premise

The proof of part (2) of Theorem 1.2 assumes, without adaptation, that an inequality previously proved for ergodic measures also holds for all invariant measures; if that extension is false, the claim that the new rate-distortion entropies equal Kolmogorov–Sinai entropy for non-ergodic measures loses its support.

Editorial extensions

If this is right

For every invariant measure on the Hilbert cube shift, partition-based and quantizer-based notions of information dimension agree, so non-ergodic measures no longer require special treatment.
The four new rate-distortion entropies give information-theoretic characterizations of Kolmogorov–Sinai entropy, meaning rate-distortion theory and classical entropy coincide in these settings.
Under the marker property with finite mean dimension, the supremum in the double variational principle can always be restricted to ergodic measures, sharpening the previously known variational formulation.
The equality extends to a wide menu of measure-theoretic ε-entropies, indicating that the choice of which ε-entropy one uses does not change the metric mean dimension for these systems.
Systems with vanishing Kolmogorov–Sinai entropy, such as translations of compact groups, automatically have zero rate-distortion entropies of all four new types.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The equality in Theorem 1.1 likely extends to shifts over arbitrary compact metric alphabets, since the proof only uses rate-distortion comparisons and the shift structure; this could be checked directly for product measures on X^Z.
The same proof technique may yield a shorter route to the known ergodic case and clarify when the lower and upper versions coincide or differ.
If the cited inequality for non-ergodic measures used in Theorem 1.2(2) is not reproved in that generality, the non-ergodic part of that theorem rests on an unverified extension; a counterexample would leave only the ergodic statement intact.
The marker-property result suggests that the maximality of ergodic measures for metric mean dimension is a general phenomenon under finite mean dimension, rather than an exceptional one.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Punchline first: the paper answers a genuinely open question from Gutman–Śpiewak, and for that result alone it should be read carefully. Theorem 1.1 extends the equality between mean Rényi information dimension and information dimension rate from ergodic to all invariant measures on the Hilbert cube shift. The proof is clean and self-contained modulo Geiger–Koch and the GS20 comparison inequality, and Lemma 3.1 is the right new tool. That is a real result, not a repackaging.

The paper also introduces four rate-distortion entropies and shows they coincide with Kolmogorov–Sinai entropy under the g-almost product property, and it widens the double variational principle to ergodic measures for a large candidate family. That is useful synthesis. The candidate set E is broad, and the Fatou argument in Lemma 3.7 is a neat way to reduce the supremum to ergodic measures.

Now the soft spot, and the reader's report is right about it. In Step 1 of Theorem 1.2 the proof invokes [W21, Proposition 4.2] for all invariant measures. That proposition is stated for ergodic measures. Footnote 4 says 'the proof applies to invariant measures' but no adaptation is shown. Ergodicity can matter in the local-entropy estimates, so the assertion is not obviously harmless. The chain in (3.7) feeding Lemma 3.4(2) rests on it. If the extension is false, the non-ergodic equality in Theorem 1.2(2) is unsupported. I could not verify the extension from the paper, and this is the main thing a referee must pin down. The theorem should be treated as conditional rather than fully proven. Theorem 1.1 does not depend on this shaky step.

The secondary concern—Lemma 2.3(3) quoted from the author's own [YCZ25]—is weaker. It is a published external result, the citation is explicit, and self-citation is not itself a flaw. Theorem 1.3 does depend on it, but that is a normal dependence on prior work, not an internal gap.

Who should read this: anyone working on mean dimension, rate-distortion theory, or measure-theoretic ε-entropies. The main theorem is solid; the rest is valuable but needs a supporting proof. Yes, send it to peer review. Ask for the invariant-measure version of [W21, Prop. 4.2] to be written out, or restrict Theorem 1.2(2) to the ergodic case if the extension fails. If that piece holds, the paper is a solid contribution.

Referee Report

3 major / 5 minor

Summary. The paper revisits the connection between mean dimension and rate-distortion theory. Theorem 1.1 proves that for every invariant measure on the Hilbert cube shift ([0,1]^Z, σ, d_Z), the lower/upper mean Rényi information dimension equals the lower/upper information dimension rate, answering a question of Gutman and Śpiewak. Theorem 1.2 introduces L_p, L∞, Bowen, and r rate-distortion entropies and proves their coincidence with Kolmogorov–Sinai entropy, for all ergodic measures and, under the g-almost product property, for all invariant measures. Theorem 1.3 establishes, for systems with the marker property and finite mean dimension, a double variational principle in which the supremum is taken over ergodic measures, for a family E of measure-theoretic ε-entropies and for L_p rate-distortion functions. The final section discusses applications to lower Brin–Katok entropy and to exchanging limsup and supremum.

Significance. If correct, the paper resolves a known open problem and gives a systematic comparison of rate-distortion entropies with classical entropy, together with an ergodic version of the double variational principle. The proof of Theorem 1.1 is mostly self-contained modulo standard external results and is clearly presented, as is the combinatorial comparison between Bowen and mean metrics in Theorem 1.2. However, the non-ergodic part of Theorem 1.2(2) relies on an unproved extension of a result of T. Wang, and Theorem 1.3 depends heavily on Lemma 2.3(3) imported from the author's previous work, so the incremental contribution is partly based on black-boxed ingredients.

major comments (3)

[§3.2, Step 1 (just after Eq. (3.8))] The proof of Theorem 1.2(2) for non-ergodic measures hinges on applying [W21, Proposition 4.2] to every invariant measure. Footnote 4 states that the proposition is proved for ergodic measures but 'the proof applies to invariant measures,' without giving the adaptation. This is load-bearing: inequality (3.7) is the only lower bound relating Pfister–Sullivan ε-entropy to the L_p rate-distortion entropy, and Lemma 3.4(2) supplies the matching upper bound only under the g-almost product property. If the asserted extension fails, the equalities in Theorem 1.2(2) are unsupported for non-ergodic μ. Please provide a full proof of the extension or explicitly restrict the statement to ergodic measures.
[§2.3, Lemma 2.3(3)] Lemma 2.3(3), quoted from the author's own [YCZ25], is used as a black box at decisive points: Lemma 3.7(1), Theorem 1.3 Step 1, and Theorem 1.3 Step 2. It contains exactly the candidate-independence and variational-principle statements that the paper claims to revisit. Because of this dependence, the manuscript should either reproduce a proof, give precise theorem numbers and hypotheses from [YCZ25], or state clearly that Theorem 1.3 is a corollary of [YCZ25, Theorems 1.1–1.3] plus the ergodic-restriction argument. As written, the novelty boundary between this paper and [YCZ25] is difficult to assess.
[§3.3, Step 1 of Theorem 1.3] The proof of Step 1 is written for h_μ(T,ε) ∈ E, but the theorem claims the inequality for h_μ(T,ε) ∈ E ∪ {R_{μ,L_p}}. The inclusion of R_{μ,L_p} is not justified in Step 1; it appears only in Step 2 via a brief '≤' remark. The missing argument is the elementary inequality R_{μ,L_1}(ε) ≤ R_{μ,L_p}(ε) for p ≥ 1, which would give sup_E rdim_{L1}(d) ≤ sup_E rdim_{Lp}(d). Please add this explicitly or adjust the theorem statement accordingly.

minor comments (5)

[§1, Introduction] Typo: 'four types pf rate-distortion entropies' should be 'four types of rate-distortion entropies'.
[§3.2] The same symbol d_n is used for the Bowen metric and the mean metric. Using \bar d_n consistently for the mean metric would remove ambiguity in the proof of (3.8).
[§3.2, Step 2] The displayed equality lim_{r→0} h_{μ,r}(T) = sup_{r>0} sup_{ε>0} R_{μ,r}(ε) = lim_{ε→0} lim_{r→0} R_{μ,r}(ε) involves an interchange of limits that should be justified by the monotonicity of R_{μ,r}(ε) in ε and r, since it is not immediate.
[§2.3, Lemma 2.3(3)] The reference to [YCZ25] should name the specific theorems (e.g., Theorems 1.1–1.3) in the bibliography entry, since Lemma 2.3(3) packages several substantial statements whose exact hypotheses matter for the present paper.
[§3.4, Theorem 3.9] Typo: 'there exists exists ε_0' should be 'there exists ε_0'.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity; central derivations are independent, with a minor self-citation and a non-circular proof gap in an external-citation extension.

full rationale

I walked the three proofs. Theorem 1.1 is not circular: after (3.1) identifies d(μ) and d̄(μ) with the L2 rate-distortion dimensions via Geiger–Koch and [GS20, Prop. C-B.1], Lemma 3.1 gives d ≤ MRID and the partition argument (3.6) gives MRID ≤ d; both inequalities are proven from the definitions rather than assumed. Theorem 1.2 is a chain of inequalities among independently defined rate-distortion entropies and Kolmogorov–Sinai entropy; no parameter is fitted and no quantity is defined as the target. The only questionable point is Section 3.2, Step 1, footnote 4 after (3.8): 'Although the statement is given for ergodic measures, the proof applies to invariant measures.' This extends [W21, Prop. 4.2] to all invariant measures without showing the adaptation; that is an omitted-proof/correctness risk, not a circular step, and Theorem 1.1 is unaffected. Theorem 1.3 uses Lemma 2.3(3), cited to the author's [YCZ25]; this is a published parameter-free theorem whose assumptions do not include Theorem 1.3, and Theorem 1.3 adds the min-over-metrics/marker-property conclusion and the R_{μ,Lp} candidate. The central claims do not reduce to their inputs by construction, so the score is low.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No empirical parameters and no new physical or mathematical entities are introduced. The central claim rests on standard ergodic theory and on two external theorems: the author's [YCZ25] classification of ε-entropy rates, and [W21, Proposition 4.2], whose extension to non-ergodic measures is asserted but not demonstrated. The latter is the most fragile pillar.

assumptions (5)

standard math Ergodic decomposition and the local entropy integral formula h_μ(T,U)=∫h_m(T,U)dτ(m)
Used in Lemma 3.7(2) to pass from invariant measures to ergodic measures.
domain assumption [YCZ25, Theorems 1.1-1.3]: for ergodic μ, limsup h_μ(T,ε)/log(1/ε) is independent of h∈E and satisfies the metric mean dimension variational principle
Self-cited black box used in Lemma 2.3(3) and Theorem 1.3; not proved in this paper.
ad hoc to paper [W21, Proposition 4.2] holds for all invariant measures (asserted extension of an ergodic-measure result)
Load-bearing for Theorem 1.2(2); the paper states the proof applies to invariant measures but gives no adaptation.
standard math For every invariant μ, lim_{ε→0} inf_{diam α≤ε} h_μ(T,α) = h_μ(T)
Used in Theorem 1.2, Step 3 to bound Bowen and r rate-distortion entropies above by Kolmogorov–Sinai entropy.
domain assumption Marker property implies existence of d∈D'(X) with mdim_M(T,X,d)=mdim(T,X) ([LT19])
Used in Theorem 1.3 Step 2 to identify the min over D'.

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Pith. "Pith review of Mean dimension and rate-distortion function revisited." pith.science (2026). https://pith.science/paper/CDVSX7AK

@misc{pith2026251008051,
  author       = {Pith},
  title        = {Pith review of: Mean dimension and rate-distortion function revisited},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CDVSX7AK}},
  note         = {Machine review of arXiv:2510.08051}
}

abstract

Around the mean dimensions and rate-distortion functions, using some tools from local entropy theory this paper establishes the following main results: $(1)$ We prove that for non-ergodic measures associated with almost sure processes, the mean R\'enyi information dimension coincides with the information dimension rate. This answers a question posed by Gutman and \'Spiewak (in Around the variational principle for metric mean dimension, \emph{Studia Math.} \textbf{261}(2021) 345-360). $(2)$ We introduce four types of rate-distortion entropies and establish their relation with Kolmogorov-Sinai entropy. $(3)$ We show that for systems with the marker property, if the mean dimension is finite, then the supremum in Lindenstrauss-Tsukamoto's double variational principle can be taken over the set of ergodic measures. Additionally, the double variational principle holds for various other measure-theoretic $\epsilon$-entropies.

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Reference graph

Works this paper leans on

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Pith tools

Reviewed August 4, 2026 · model on record in the stance chip above.

[1] [1]

Adler, A

R. Adler, A. Konheim and M. McAndrew, Topological entropy, Trans. Amer. Math. Soc. 114 (1965), 309-319

1965

[2] [2]

Brin and A

M. Brin and A. Katok, On local entropy , Geometric dynamics (Rio de Janeiro), Lecture Notes in Mathematics, Springer, Berlin 1007 (1983), 30-38

1983

[3] [3]

Carvalho, G

M. Carvalho, G. Pessil and P. Varandas, A convex analysis approach to the metric mean dimension: limits of scaled pressures and variational principles , Adv. Math. 436 (2024), Paper No. 109407, 54 pp

2024

[4] [4]

T. M. Cover and J. A. Thomas, Elements of information theory, second edition, Wiley, New York, 2006

2006

[5] [5]

Denker, C

M. Denker, C. Grillenberger and K. Sigmund, Ergodic theory on compact spaces , Springer-Verlag, Berlin-New York, 1976

1976

[6] [6]

Geiger and T

B. Geiger and T. Koch, On the information dimension rate of stochastic processes, IEEE International Symposium on Information Theory (ISIT). IEEE, 2017: 888-892

2017

[7] [7]

Geiger and T

B. Geiger and T. Koch, On the information dimension of stochastic processes, IEEE Trans. Inform. Theory 65 (2019), 6496-6518

2019

[8] [8]

Gray, Entropy and information theory, Springer, New York, Second edition, 2011

R. Gray, Entropy and information theory, Springer, New York, Second edition, 2011

2011