Entropic Symmetrization Resistance

Emma Pollard; Mokshay Madiman

REVIEW 3 major objections 6 minor 12 references

For any lopsided coin flip X, every independent symmetrizer Y satisfying X+Y symmetric must have entropy at least H(X).

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2026-08-03 15:11 UTC pith:P5AUVPWX

load-bearing objection Entropic symmetrization resistance for asymmetric Bernoulli is a genuine new result with a small, patchable gap in the main proof; referee it. the 3 major comments →

arxiv 2607.29020 v1 pith:P5AUVPWX submitted 2026-07-31 math.PR cs.ITmath.IT

Entropic Symmetrization Resistance

Mokshay Madiman , Emma Pollard This is my paper

classification math.PR cs.ITmath.IT MSC 60C05

keywords entropic symmetrization resistanceasymmetric BernoulliShannon entropysymmetrizersvariance symmetrizationcompact groupsfinite cyclic groupsaffine basis

verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

An asymmetric Bernoulli random variable—a two-outcome coin flip with probabilities p and 1−p, p≠1/2—cannot be entropically 'improved' by symmetrization. The paper proves that if Y is independent of X and X+Y has a distribution symmetric about zero, then the entropy H(Y) is at least H(X), with equality exactly when Y is the reflection of an independent copy of X. This is the entropy analogue of an earlier variance result, and it matters because entropy, unlike variance, is defined on any locally compact group, so the question can be asked far beyond the real line. On compact groups the paper shows a strong boundary effect: any density that stays bounded away from zero cannot be entropically symmetrization resistant, and it explicitly classifies the resistant distributions on Z3 and Z4.

Core claim

The paper's central claim is that for X distributed as Bernoulli(p,a,b) with p≠1/2, every independent symmetrizer Y in R satisfies H(Y) ≥ H(X), with equality if and only if Y has the distribution of −X. The proof reduces to X supported on {−1,1} with p>1/2, shows that any candidate counterexample can be projected onto the even sublattice 2Z without increasing entropy, and then uses an affine-basis expansion of the symmetrizer space to obtain a contradiction: the coefficient at 0 would have to be simultaneously greater than 1 and at most 1. The paper also proves a d-dimensional analogue with constant 1/d, a variance analogue for nonnegative random variables, and, for compact groups, that posi

What carries the argument

The central object is an affine basis for the space Sym of all symmetrizer probability mass functions of an asymmetric Bernoulli. The paper constructs atomic symmetrizers ``\hat f_k^r``—each a mixture of a reflected copy of X shifted to a pair of opposite points—and proves that every symmetrizer expands as a sum of these atoms with coefficients summing to 1. A companion lemma shows that any negative coefficient is dominated in magnitude by its positive neighbors. These two facts let the proof turn the entropy comparison into a one-coefficient contradiction, and they also drive the new variance proof and the multidimensional result.

Load-bearing premise

The proof rests on the claim that every symmetrizer of an asymmetric Bernoulli expands as an infinite weighted sum of the paper's special two-point symmetrizers, with every negative weight dominated by larger positive neighboring weights; if that expansion or the coefficient-sign lemma fails for symmetrizers with infinitely many support points, the entropy lower bound is unproved.

What would settle it

Search for a finite-support symmetrizer f of X~Bernoulli(0.9, −1, 1) satisfying the symmetry equations q f(z+2)+p f(z)=q f(−z)+p f(−z−2) for all z; if any such f has H(f)<H(0.9), Theorem 4.8 is false. More directly, test Lemma 4.7 by computing the coefficients α_j from its explicit formula and checking whether an infinite-support sequence with a negative α_j can still be nonnegative when its neighbors are smaller than (p/q)|α_j|—finding one would break the proof even if the theorem is true.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

For a d-dimensional product of asymmetric Bernoulli coordinates, any independent symmetrizer Y satisfies H(Y) ≥ (1/d) H(X); if the coordinates of Y are required to be independent, the stronger bound H(Y) ≥ H(X) holds.
On any compact group, a probability density with positive infimum cannot be entropically symmetrization resistant, so resistant distributions must lie on the boundary of the probability simplex.
The entropically symmetrization resistant distributions on Z3 are exactly the extreme two-thirds of each side of the simplex; on Z4 they are exactly the two-point distributions supported on adjacent pairs not contained in a coset of {0,2}.
Every probability measure on the positive real line is variance symmetrization resistant with an explicit constant; for example, the exponential distribution has constant approximately 0.47.
Minimum-entropy symmetrizers need not be unique: the paper exhibits resistant distributions on Z3 and Z4q, with q odd, that have two distinct entropy-minimizing symmetrizers.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

My inference: If the affine-basis proof is robust, the same method may test Rényi entropies of other orders; a failure for some order would indicate that resistance is a distinctly Shannon-entropy phenomenon rather than a general property of spread functionals.
My inference: The paper's auxiliary quantity, minimum mutual information between X and X+Y subject to symmetry, suggests a concrete communication interpretation: any scheme that symmetrizes by adding independent noise must pay at least H(X) bits of coupling, which could connect to rate limits in the capacity problems the paper mentions.
My inference: The compact-group boundary theorem suggests entropic symmetrization resistance is inherently an extreme-point property; a testable conjecture is that on larger cyclic groups, all resistant distributions have very small support, possibly only two or three points.
My inference: The d-dimensional factor 1/d appears loose when the coordinates of Y are dependent; the equality conditions in the theorem suggest that constructing a totally dependent symmetrizer with H(Y)=H(Y_i) would sharpen the constant, an avenue the paper leaves open.

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

Entropic symmetrization resistance for asymmetric Bernoulli is a genuine new result with a small, patchable gap in the main proof; referee it.

read the letter

Short version: the entropic symmetrization resistance theorem for asymmetric Bernoullis is real, new, and essentially correct; the main proof has one small gap that is patchable, and the stress-test pointed at the right spot. I would send this to a referee.

What is new: Kagan et al. and Pal proved variance symmetrization resistance for asymmetric Bernoulli. This paper adds the entropy version (Theorem 4.8), gives multidimensional entropy and covariance bounds (Theorems 4.11 and 4.17), and classifies entropic symmetrization resistant distributions on Z3 and Z4, plus a clean compact-group boundary theorem: if a density has positive infimum, it is not resistant. The affine-basis expansion for symmetrizer PMFs is a genuinely useful technique, and the paper is honest about prior work—it cites Kagan and Pal, and notes Jiange Li's confirmation of the exponential conjecture. I found no circularity, no parameter fitting, and no misleading claims.

Where the soft spots are: - In Theorem 4.8, the line "by Lemma 4.7, ∑_{i≥2} α_i ≥ 0" is asserted, not proved. The stress-test is right that this is load-bearing: the contradiction α1 > 2p > 1 versus α1 ≤ 1 depends on it. It is also patchable. For α_j ≤ 0, Lemma 4.7 gives α_{j+1} ≥ (p/q)|α_j|, and those right neighbors are distinct for distinct negative j, so the negative part of the tail is dominated by the positive part. The authors should add that sentence. - The basis expansion in Lemmas 2.17 and 2.20 interchanges infinite sums without explicitly stating absolute convergence. The coefficient formula gives ∑|α_i| ≤ 2/(p−q) < ∞, so this is minor. Also, the r∈(0,1) case is less central to Theorem 4.8 than the stress-test suggests: the reduction to Sym0 means Lemma 2.17 is what the entropy theorem needs, and the proof actually does the computation for one representative point. - Section 3.2 applies optional stopping with regularity assumptions left implicit. Again minor; Skorokhod embedding with finite variance gives finite expected stopping time, and the non-smooth ρ is covered by the cited Aebi extension.

The finite-group sections look careful; the Z4 case analysis is long but explicit. I do not see a problem that would sink the paper. If I were refereeing, I would ask for the tail-sum argument in Theorem 4.8 and a sentence on absolute convergence, then accept.

This is for people who work on entropies, symmetrization, or convolution inequalities on groups. A serious referee should take it.

Referee Report

3 major / 6 minor

Summary. The paper introduces entropic symmetrization resistance for probability measures on locally compact groups, as an entropy analogue of the variance-based notion of Kagan, Mallows, Shepp, Vanderbei and Vardi. The main result (Theorem 4.8) asserts that an asymmetric two-point Bernoulli random variable X satisfies H(Y) >= H(X) for every independent symmetrizer Y, with equality only for Y distributed as -X. The proof proceeds by analyzing the affine structure of the space of symmetrizer probability mass functions, writing symmetrizers supported on the even coset as infinite signed combinations of elementary symmetrizers. The paper also proves a multidimensional entropic bound on {±1}^d, a covariance-matrix analogue, a new proof and extension of the variance result to nonnegative variables, and, for compact groups, proves that densities with positive infimum cannot be entropically symmetrization resistant and gives exact classifications of resistant distributions on Z_3 and Z_4.

Significance. If the main theorem is correct, this is the first entropy analogue of the 1999 variance result, and the compact-group boundary theorem and exact Z_3/Z_4 classifications are new and clean. The paper is self-contained and does not fit parameters or tailor normalizations to force the conclusion; prior work by Kagan et al. and Pal is used for context and comparison. The multidimensional statements are natural and the proof strategy via the affine basis of the symmetrizer space is interesting. However, the central argument relies on infinite sums of signed coefficients and on several coefficient-sign assertions that are not fully justified as written. These gaps are localized and appear repairable, but they must be fixed before the main claim is fully established.

major comments (3)

[§4.1, Theorem 4.8] The line 'by Lemma 4.7, sum_{i>=2} alpha_i >= 0' is not a direct consequence of Lemma 4.7 as stated. Lemma 4.7 gives pairwise neighbour inequalities for each negative coefficient, but one must explicitly show that negative coefficients are non-adjacent and pair each negative alpha_j with alpha_{j+1}; then alpha_j+alpha_{j+1} >= (p/q - 1)|alpha_j| >= 0, so the tail sum is nonnegative. Without this argument the contradiction alpha_1 > 2p > 1 versus alpha_1 <= 1 is unproved. The same missing pairing argument appears in Lemma 3.3, where alpha_0^r+alpha_1^r <= 1 is inferred from Lemma 2.22; a single general tail-sum lemma would cover both places.
[§2.3, Lemmas 2.17, 2.20, 2.21] The basis expansions use infinite sums of signed coefficients without any convergence or domination argument. For r=0 this is easily fixed: alpha_i = 2(p f(-2i)-q f(2i))/(p-q), so sum_i |alpha_i| <= 2/(p-q) < infinity, which justifies interchanging sums and the identity sum alpha_i = 1. The proof should state this. For Lemma 2.20 with r in (0,1), the displayed computation only covers z=2n+r; the case z=2n-r is dismissed as 'almost identical', and no argument is given for convergence of the bi-infinite series or uniqueness of the coefficients. Since Lemma 2.20 feeds Theorem 2.21, Lemma 2.22 and the variance proof, this is not a cosmetic omission.
[§4.1, Lemma 4.6] The proof states 'Noting that h(0)=0, Lemma 4.4 shows that H(h)>H(X)'. Lemma 4.4 is stated for |supp|>=3; if |supp(h)|=2, Lemma 4.1 forces h=f_{-X}, giving H(h)=H(X). In that case the strict inequality H(h)>H(X) is false. The lemma can be repaired by observing that strict concavity of entropy still gives H(f)>cH(g)+(1-c)H(h) when f=cg+(1-c)h and g is distinct from h, so the desired conclusion survives after a short case discussion. As written, however, the proof is incomplete at a step used in the reduction to Sym_0.

minor comments (6)

[§1, Definition 1.1] The definition of a spread functional requires upper semicontinuity, but neither variance nor Shannon entropy is upper semicontinuous in the topology of weak convergence. Please clarify what topology or extension is intended, or weaken this property for the paper's purposes.
[§3.2, Theorem 3.5] The application of optional sampling to obtain E M_T = 0 should be justified explicitly: T has finite mean by the Skorokhod embedding and rho'' is bounded, so the standard optional stopping theorem applies. A one-sentence remark would suffice.
[§2.3, Lemma 2.20] There is a duplicated 'the' in the sentence 'where we have used the the symmetry equation'.
[§2.2, Remark 2.14] The identity 1 = (1/(n+1)) H(V) appears inconsistent with V uniform on [-2n,2n]\{0}, whose entropy is log_2(4n). If the intended support is [-2^n,2^n]\{0}, please write it accordingly.
[§4.2, Theorem 4.17] In equality condition (iii), the phrase 'for all i in [d]' is redundant and should be removed; J_d is already the matrix in question.
[§5.5, Example 5.15] The final numerical interval (0.46075, 0.53925) is presented without explanation of the computation. If this is an approximate numerical result, say so; if it is exact, provide the expression or the code used.

Circularity Check

0 steps flagged

No significant circularity: the entropy symmetrization theorem is derived from in-paper algebra, not from fitted inputs or self-cited uniqueness.

full rationale

The central claims are self-contained. Theorem 4.8's proof is a contradiction argument built from the Bernoulli convolution/symmetry equations: the affine-basis expansion of Sym0 (Lemma 2.17), the coefficient-sign domination (Lemma 4.7), and the reduction of a hypothetical counterexample to Sym0 (Lemma 4.6) are all established in the paper from the definition of Sym, without invoking the variance result or any prior work as the source of the entropy bound. No parameter is fitted to the target inequality, no normalization is chosen to force the conclusion, and no load-bearing uniqueness theorem is imported from the authors' earlier papers; citations to Kagan et al., Pal, and the authors' own articles are contextual or alternative-proof references. The one unexpanded inference in Theorem 4.8 ('by Lemma 4.7, ∑_{i≥2}α_i≥0') is a patchable proof gap, not a circular reduction: it follows by pairing negative coefficients with their positive neighbors and does not assume the theorem being proved. The compact-group boundary theorem (5.3) likewise follows directly from convexity and the linearity of the symmetry equations.

Axiom & Free-Parameter Ledger

0 free parameters · 4 axioms · 0 invented entities

No free parameters are fitted; constants such as c for exponential or Poisson are computed from the given distributions. The paper relies only on standard probability/information-theoretic tools plus one unstated regularity assumption in the stochastic-calculus section. No new physical or mathematical entities are postulated.

axioms (4)

standard math Skorokhod embedding and Aebi's extended Ito formula for non-C^2 functions
Used in Theorem 3.5 to bound Var(Y) via stochastic calculus; standard but external to the paper.
domain assumption Optional stopping holds for the martingale and stopping time T without explicit integrability verification
Invoked silently in the proof of Theorem 3.5; a standard but nontrivial regularity condition.
standard math Shannon entropy is strictly concave and translation invariant for discrete and compact-group distributions
Used in Lemma 2.13 and Theorem 5.3 to compare entropies of mixtures and convolutions.
standard math Haar measure on compact groups exists and the uniform density is a symmetrizer of any density
Used in Lemma 5.2 and Theorem 5.3 to construct the uniform symmetrizer.

pith-pipeline@v1.3.0-daily-deepseek · 27843 in / 14520 out tokens · 147399 ms · 2026-08-03T15:11:13.283104+00:00 · methodology

0 comments

read the original abstract

An asymmetric random variable $X$ in the reals is said to be variance symmetrization resistant if every independent random variable $Y$ in the reals that produces a symmetric sum $X+Y$ has a greater variance than that of $X$. Asymmetric Bernoulli random variables were shown to be variance symmetrization resistant by Kagan, Mallows, Shepp, Vanderbei, and Vardi (1999); Pal (2008) gave a proof using stochastic calculus. We introduce the notion of entropic symmetrization resistance on locally compact groups-- this means that the entropy of any independent symmetrizer $Y$ must exceed that of $X$. We show that asymmetric Bernoulli random variables exhibit entropic symmetrization resistance, and show a multidimensional generalization. We also explore basic aspects of the entropic symmetrization resistance problem in compact groups. In particular, we show that any distribution on a finite group that is entropic symmetrization resistant must lie on the boundary of the probability simplex, and describe precisely the class of all entropic symmetrization resistant distributions on $\mathbb{Z}_3$ and $\mathbb{Z}_4$.

Figures

Figures reproduced from arXiv: 2607.29020 by Emma Pollard, Mokshay Madiman.

**Figure 1.** Figure 1: The probability simplex for Z3. The red parts of the simplex are symmetrization resistant. A representative symmetrization-resistant PMF f with symmetrizer space Sym(f) is shown. these points is f−; denote the other by p. Thus for g ∈ Sym(f) = conv(f−, p) there exists λ ∈ [0, 1] such that g = λp + (1 − λ)f− and H(g) ≥ λH(p) + (1 − λ)H(f−), In turn, H(f−) = H(f) and f is not symmetrization resistant if and … view at source ↗

**Figure 2.** Figure 2: The facet of the probability simplex for [PITH_FULL_IMAGE:figures/full_fig_p028_2.png] view at source ↗

**Figure 3.** Figure 3: The probability simplex ∆4. Symmetrization resistant distributions in Z4 are represented in red (Theorem 5.12). Page 28/32 [PITH_FULL_IMAGE:figures/full_fig_p028_3.png] view at source ↗

discussion (0)

Reference graph

Works this paper leans on

12 extracted references

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Madiman, P

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Itô’s formula for non-smooth functions

Robert Aebi. Itô’s formula for non-smooth functions. Publications of the Research Institute for Mathematical Sciences, 28(4):595–602, 1992

1992

[2] [2]

Eskenazis, P

A. Eskenazis, P . Nayar, and T. Tkocz. Gaussian mixtures: entropy and geometric inequalities. Ann. Probab., 46(5):2908–2945, 2018

2018

[3] [3]

Horn and Charles R

Roger A. Horn and Charles R. Johnson. Matrix analysis. Cambridge University Press, 2nd edition, 2012

2012

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Kagan, C

A. Kagan, C. L. Mallows, L. A. Shepp, R. J. Vanderbei, and Y. Vardi. Symmetrization of binary random variables. Bernoulli, 5(6):1013–1020, 1999

1999

[5] [5]

How much randomness is needed to symmetrize a random variable?, 2026

Jiange Li. How much randomness is needed to symmetrize a random variable?, 2026

2026

[6] [6]

Madiman, P

M. Madiman, P . Nayar, and T. Tkocz. On the question of the best additive noise among symmetric log-concave noises. In Proc. IEEE Intl. Symp. Inform. Theory., 2019

2019

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Madiman, P

M. Madiman, P . Nayar, and T. Tkocz. Sharp moment-entropy inequalities and capacity bounds for log-concave distributions. IEEE Trans. Inform. Theory, 67(1):81–94, January 2021

2021

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S. Pal. Symmetrization of Bernoulli. Electron. Commun. Probab., 13:194–197, 2008

2008

[9] [9]

A study of some entropy inequalities

Emma Pollard. A study of some entropy inequalities . PhD thesis, University of Delaware, 2023

2023

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Information theory: from coding to learning

Yury Polyanskiy and Yihong Wu. Information theory: from coding to learning . Cambridge University Press, 2025. Page 31/32 Entropic Symmetrization Resistance

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C.E. Shannon. A mathematical theory of communication. Bell System Tech. J., 27:379–423, 623–656, 1948

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Measures of symmetry for convex sets and stability, volume 776

Gabor Toth. Measures of symmetry for convex sets and stability, volume 776. Springer, 2015. Page 32/32

2015