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Near-Gaussian counterexamples to the Ball-Nayar-Tkocz entropy concavity conjecture

Congyi Luo

arXiv:2609.11418v2cs.ITmath.ITmath.PR

TL;DR

The paper asks whether entropy concavity for Gaussian inputs extends to nearby densities or follows from fixed Gaussian smoothing. It constructs strongly log-concave counterexamples using an endpoint expansion for asymmetric inputs and high-order Hermite perturbations for symmetric inputs. The counterexamples persist for every fixed q > 0 in the asymmetric case and for q > 4/7 in the symmetric case, while approaching the Gaussian.

  • Problem

    The paper asks whether the Ball-Nayar-Tkocz entropy concavity conjecture holds near the Gaussian and whether fixed Gaussian smoothing suffices to ensure concavity.

    Source: §1 · p. 1

  • Method

    The asymmetric construction uses an endpoint entropy expansion, while the symmetric construction compares the decay of high-order Hermite perturbations under different weights.

    Source: §1 · p. 1

  • Results

    Asymmetric counterexamples exist for every fixed 0 < q ≤1, while symmetric counterexamples exist for every 4/7 < q ≤1.

    Source: §1 · p. 2, §1.1 · pp. 2–3

  • Takeaways & Limitations

    Both families remain strongly log-concave and approach the Gaussian in relative density and fixed-order derivatives; symmetric examples can also match any prescribed finite number of Gaussian moments exactly.

    Source: §1 · p. 2

    Whether uniform concavity holds for symmetric inputs when 0 < q ≤4/7 remains unresolved.

    Source: §9 · p. 15, §10.2 · p. 15

Abstract

from arXiv · show

The Ball-Nayar-Tkocz conjecture asserts that, for independent real random variables $X,Y$ with a common log-concave density, the function $t\mapsto h(\sqrt tX+\sqrt{1-t}Y)$ is concave on $[0,1]$. Gaussian distributions have this property. We construct strongly log-concave counterexamples arbitrarily close to the Gaussian and study their persistence under Gaussian smoothing. Let $X$ have mean zero and variance one, let $G$ be an independent standard Gaussian, and let $q\in[0,1]$ denote the signal variance proportion in the smoothed variable $\sqrt qX+\sqrt{1-q}G$. In the asymmetric case, for every fixed $0<q\le1$, there are counterexamples for which the entropy of the weighted sum of two independent copies of the smoothed variable has strictly positive second derivative near the endpoints. In the symmetric case, for every $4/7<q\le1$, there are counterexamples for which an unequally weighted sum has strictly greater entropy than the equally weighted sum. Both families have smooth, strictly positive densities and are strongly log-concave before and after smoothing. The density ratios $f/\varphi$ converge uniformly to one, and each derivative of fixed positive order converges uniformly to zero, where $\varphi$ is the standard Gaussian density. The symmetric counterexamples can also match any prescribed finite number of Gaussian moments exactly. The asymmetric construction uses an endpoint expansion of entropy, whereas the symmetric construction exploits the different decay rates of high-order Hermite perturbations under different weights. Thus, densities violating concavity approach the Gaussian density in the strong sense described above. The question of uniform concavity in the symmetric class for $0<q\le4/7$ remains open.

1. Introduction and main theorems

The paper asks whether Gaussian inputs’ entropy concavity extends near the Gaussian or survives fixed Gaussian smoothing, and constructs counterexamples in asymmetric and symmetric classes. These counterexamples remain strongly log-concave and can approach the Gaussian with strong uniform approximation.

1. Introduction and main theorems

  • The conjecture concerns concavity of the entropy of a variance-normalized weighted sum of independent variables with a common log-concave density; for Gaussian inputs, this entropy curve is constant.The paper asks whether concavity holds near the Gaussian and whether fixed Gaussian smoothing suffices.Source: §1 · p. 1

1.1. Main theorems.

  • Both counterexample families remain strongly log-concave before and after smoothing while their relative densities converge uniformly to the Gaussian and fixed-order derivatives converge to zero.The symmetric construction can additionally match any prescribed finite number of Gaussian moments exactly.Source: §1 · p. 2, §1.1 · p. 2
  • For every fixed 0 < q ≤1, asymmetric counterexamples have strictly convex entropy curves near an endpoint; symmetric counterexamples violate the midpoint maximum for every 4/7 < q ≤1.The asymmetric sequence may depend on q; the symmetric comparison is between an unequal weight and the equal weight.Source: §1.1 · p. 2

1.2. Related results and the proofs.

  • The symmetric counterexamples show that equal weights need not maximize entropy in the strongly log-concave class, unlike the stated results for Gaussian scale mixtures [5, Theorem 8], [4, Theorem 1].The paper fixes the squared sum of the weights, preserving the variance of the weighted sum.Source: §1.2 · p. 3, §1.1 · p. 3
  • The asymmetric construction uses an endpoint expansion, while the symmetric construction compares decay rates of high-degree Hermite perturbations at midpoint and nonmidpoint weights.In the asymmetric case, choosing opposite signs for the third central moment and the third logarithmic-derivative moment yields positive endpoint curvature.Source: §1.2 · pp. 3–4

2. Gaussian smoothing and Hermite functions

This section develops tools that turn entropy comparisons into estimates of relative-density perturbations and describe how Gaussian smoothing acts on them. Hermite functions provide an explicit handle on the smoothed perturbations and their norms.

2. Gaussian smoothing and Hermite functions

  • Entropy comparisons at equal variance reduce to Gaussian L2-norm comparisons of relative perturbations, using two-sided quadratic bounds for relative entropy.The bounds apply when the density ratio to the Gaussian is uniformly bounded near one.Source: §2 · p. 4
  • A generating-function identity gives exact degree-dependent L2 norms for smoothed Hermite functions, while derivative bounds support positivity and strong log-concavity of small perturbations.The norm identity is derived by integrating Mehler’s formula.Source: §2 · pp. 4–6
  • Gaussian smoothing maps a relative perturbation g to T_ρg, is self-adjoint on L2(φ), and sends each Hermite polynomial H_j to ρ^jH_j.For the smoothed variable in the paper, ρ = √q.Source: §2 · p. 5
  • The smoothing operator contracts L2(φ) and L∞ norms, and its mth derivative is bounded by ρ^m times the corresponding derivative norm.These bounds help control perturbations and their derivatives under smoothing.Source: §2 · pp. 4–5

3. Counterexamples for asymmetric inputs

For any fixed positive smoothing proportion, the asymmetric construction uses small smooth perturbations to produce positive entropy curvature near an endpoint. It separately adjusts two third-moment quantities to meet the endpoint expansion’s sign condition.

3. Counterexamples for asymmetric inputs

  • For every fixed q ∈(0, 1], arbitrarily small smooth strongly log-concave asymmetric perturbations yield strictly positive entropy curvature near an endpoint after smoothing.The construction changes amplitudes of two fixed functions rather than increasing an oscillation degree.Source: §3 · pp. 6, 8
  • Choosing the smoothed density’s third central moment positive and its third logarithmic-derivative moment negative makes their product negative, which gives strict endpoint convexity.The endpoint expansion supplies this sufficient sign condition.Source: §3 · p. 6
  • Fourth- and fifth-order Hermite components preserve the first three moments while producing the needed logarithmic-derivative contribution; a smaller third-order component adjusts the third central moment.The construction has the form f_ε = φ(1 + εu + ε²d_qv).Source: §3 · p. 7
  • Taking perturbation amplitudes to zero makes every fixed-order derivative of the relative density converge uniformly to zero before and after smoothing.This follows from expressing the perturbations as linear combinations of fixed Schwartz functions.Source: §3 · p. 8

4. Perturbations preserving finitely many Gaussian moments

High-degree Hermite perturbations are corrected in a finite-dimensional space to match Gaussian moments while preserving positivity and strong log-concavity. Their relative densities approach the Gaussian, including after smoothing.

4. Perturbations preserving finitely many Gaussian moments

  • For any fixed K, a finite-dimensional correction of high-degree Hermite perturbations matches Gaussian moments through order 2K exactly while preserving positivity and strong log-concavity.The moment correction decays exponentially with degree.Source: §4 · pp. 8–9
  • The corrected densities converge to the Gaussian in every fixed-order relative norm, with their relative density eventually remaining positive.The zeroth- and second-moment corrections enforce unit mass and variance one, and evenness makes odd moments vanish.Source: §4 · p. 9
  • After smoothing, the log-density curvature converges uniformly to the Gaussian curvature for q in [0, 1], retaining strong log-concavity.The stated curvature convergence is uniform over the full smoothing range.Source: §4 · p. 9

5. Weighted convolution and the linear term

The weighted-convolution density is represented through conditional expectations of the two perturbations. Its relative-density expansion separates linear and quadratic terms, and the linear term admits an exponential upper bound.

5. Weighted convolution and the linear term

  • The weighted convolution is represented using a conditional expectation over three independent standard Gaussians, with the two smoothed copies contributing linear and quadratic perturbation terms.The output relative density is decomposed exactly into terms by perturbation amplitude.Source: §5 · p. 10
  • An exponential upper bound on the linear term's norm enables comparison with the midpoint quadratic term to determine when it is negligible.The norm comparison uses the larger of the two weights and the stated parameter range q >= 4/7.Source: §5 · p. 10
  • The linear terms are Gaussian-smoothed perturbations evaluated at parameters determined by the two weights.The conditional expectations are T_sqrt(qt) and T_sqrt(q(1-t)) applied to the perturbation.Source: §5 · p. 10

6. Positive coefficient comparison for the quadratic term

The quadratic perturbation is analyzed through a four-variable Gaussian integral and a generating function with positive coefficients. Comparing coefficient bounds at and away from the midpoint yields different exponential decay rates.

6. Positive coefficient comparison for the quadratic term

  • The squared norm of the quadratic term is expressed as a four-variable Gaussian integral and an exact generating-function coefficient.The Gaussian variables arise from conditionally independent copies given the weighted sum.Source: §6 · pp. 10–11
  • At a fixed nearby nonmidpoint weight, nonnegative coefficients allow an upper bound by their total-degree sum, producing a different exponential decay rate.The midpoint lower bound and nonmidpoint upper bound are the comparison used for the quadratic term.Source: §6 · pp. 10–12
  • At the midpoint, retaining one positive coefficient gives a lower bound proportional to n^-4κ^(2n) for the quadratic norm squared.The selected coefficient is bounded below using a multinomial probability estimate.Source: §6 · pp. 11–12

7. A strict entropy gap

For q>4/7, the quadratic term dominates the linear term at the midpoint, and a fixed nonmidpoint weight has strictly greater entropy. Symmetry then turns this comparison into a violation of concavity.

7. A strict entropy gap

  • For every fixed q>4/7, a fixed nonmidpoint weight has strictly greater entropy than the midpoint for sufficiently large degree.The comparison weight can be chosen near the midpoint; on any compact interval [q0, 1] with q0>4/7, a common comparison weight and degree threshold are available.Source: §7 · pp. 12–14
  • The midpoint quadratic norm is bounded below while the linear norm is bounded above, making the linear contribution negligible for q>4/7.Moment corrections do not change these comparisons.Source: §7 · pp. 12–13
  • The threshold comparison between κ^2 and τ(q/2) is strict precisely for q>4/7 in the stated interval.At q=4/7, the strict inequality does not hold.Source: §7 · pp. 12–13
  • Because exchanging the independent copies gives F(t)=F(1-t), the strict comparison at a pair of weights symmetric about the midpoint contradicts concavity.Source: §7 · p. 13

8. Consequences of the main theorems

For each K and q>4/7, the Gaussian lies on the relative boundary of the symmetric concavity and midpoint-maximization sets. The counterexamples approach it while their entropy gaps vanish, and uniform concavity in the symmetric class requires q≤4/7.

Boundary behavior

  • For every K≥1 and 4/7<q≤1, the standard Gaussian belongs to the relative boundaries of both the concavity set Cq,K and midpoint-maximization set Mq,K.The approximating counterexamples lie outside both sets while converging to the Gaussian in every fixed-order relative norm.Source: §8 · p. 14
  • The strict entropy gaps of the approximating counterexamples tend to zero as the perturbation amplitude vanishes.Uniform convergence of relative densities and the entropy convergence lemma yield convergence of the weighted-sum relative entropies.Source: §8 · p. 14

Necessary noise condition

  • Uniform concavity over symmetric smooth strongly log-concave variance-one densities at fixed q requires q≤4/7.For smoothing written as X+σG, the argument rescales the signal so q=v/(v+σ²).Source: §8 · p. 14

9. Conclusions

The paper disproves entropy concavity arbitrarily close to the Gaussian in specified asymmetric and symmetric smoothing ranges. These counterexamples remain strongly log-concave, while symmetric counterexamples for q≤4/7 remain an open question.

Main results

  • For every 0<q≤1, asymmetric counterexamples have strictly convex entropy curves near an endpoint; for every 4/7<q≤1, symmetric counterexamples give greater entropy at an unequal weight than at equal weighting.Both results concern inputs arbitrarily close to the Gaussian.Source: §9 · p. 15

Approximation properties

  • Both counterexample families remain strongly log-concave before and after smoothing, with relative densities and each fixed-order derivative converging uniformly to their Gaussian limits.The symmetric family can also match any prescribed finite number of Gaussian moments exactly.Source: §9 · p. 15

Open range

  • For symmetric inputs, whether concavity holds for 0<q≤4/7 remains unresolved; 4/7 is not established as a critical value.The paper identifies it as the boundary of an auxiliary interpolation estimate.Source: §9 · p. 15

10. Further questions

The further questions separate uniform concavity across symmetric inputs from counterexamples near the Gaussian and from local curvature at equal weighting. The unresolved range is q≤4/7, where the interpolation boundary does not itself settle concavity.

10. Further questions

  • The open problems distinguish uniform concavity, local concavity near the Gaussian, and the entropy curve’s shape near equal weighting.Answers would clarify the roles of symmetry and Gaussian smoothing in entropy concavity.Source: §10 · p. 15, §10.3 · p. 16

10.1. Is there a positive range of uniform concavity for symmetric inputs?

  • For symmetric variance-one log-concave inputs, the paper asks whether any q in (0,4/7] guarantees concavity uniformly.The value 4/7 comes from an interpolation inequality and does not determine the entropy derivative’s sign or exclude other constructions.Source: §10.1 · p. 15

10.2. Can counterexamples still be arbitrarily close to the Gaussian?

  • The paper asks whether counterexamples in the remaining range can approach the Gaussian while matching its moments through order 2K exactly.This local question is stronger than failure of uniform concavity; local concavity near the Gaussian could hold even if uniform concavity fails.Source: §10.2 · pp. 15–16

10.3. Can the midpoint entropy comparison be strengthened to strictly positive curvature?

  • The paper asks whether symmetric counterexamples can have strictly positive second derivative at equal weighting, rather than only a higher entropy at a fixed unequal weight.Positive curvature would make equal weighting a strict local minimum; the existing fixed-weight comparison does not directly establish this.Source: §10.3 · p. 16

Appendices

The appendices develop criteria and estimates for entropy counterexamples, including explicit cosine and general smooth even periodic perturbations that yield unequal-weight entropy violations. They also examine how weighted convolution produces and concentrates the entropy gap, and outline further analyses of symmetric profiles, midpoint curvature, the 4/7 bound, and parameter choices.

Appendices

  • The roadmap assigns Appendices D–G to general even periodic profiles and midpoint scaling, midpoint curvature, sharp interpolation estimates explaining 4/7, and explicit parameters with a positive entropy-gap bound.Source: Appendices · p. 17

Appendix A. Endpoint expansions for asymmetric densities

  • The leading endpoint curvature depends on the product of the third central moment and the third logarithmic-derivative moment; a negative product suffices for strict convexity near the endpoint.For even densities both moments vanish, so this criterion does not produce the symmetric counterexamples.Source: Appendix A · p. 17

Appendix B. Moment estimates for asymmetric perturbations

  • Fourth- and fifth-order Hermite components preserve the first three moments, while their even–odd cross term adjusts the logarithmic-derivative moment; a smaller third-order component adjusts the third central moment.The cross-term coefficient is negative when K > 1.Source: Appendix B · p. 20, §B.1 · p. 20

B.2. A bound for the third moment of the logarithmic derivative.

  • For the fixed perturbations, K > 1 gives J3 = 3(1 − K)ε^2 + O(ε^3), so sufficiently small positive amplitudes yield asymmetric endpoint counterexamples.The same sign condition is realized after Gaussian smoothing in the main text.Source: §B.2 · p. 22

Appendix C. An explicit family of cosine counterexamples

  • For every k ≥14, a single-cosine perturbation gives an entropy gap between weights 1/5 and 1/2 strictly greater than e^-4k/256, violating concavity.The amplitude tends to zero as frequency increases, and each fixed-order derivative tends uniformly to zero.Source: Appendix C · p. 22

C.1. Convolution formulas and entropy comparison.

  • Equal weighting retains an undamped oscillatory frequency, whereas at weight 1/5 every oscillatory term is exponentially damped; equal variances make the entropy comparison equivalent to comparing relative entropies.The appendix derives the convolution and entropy comparison formulas for the cosine densities.Source: Appendix C · p. 22
  • The inequality F_k(1/2) < F_k(1/5) + F_k(4/5) directly violates concavity.Source: §C.1.4 · p. 24

D.1. An exponential cosine potential.

  • For every k ≥ 10, the exponential-cosine construction has F_k(1/5) = F_k(4/5) > F_k(1/2) by more than 1/(25k^12).The standardized negative logarithmic curvature converges uniformly to 1, and the density and its first two derivatives converge uniformly to the corresponding Gaussian derivatives.Source: §D.1 · p. 25

D.2. Pointwise entropy deficits for arbitrary smooth even periodic waveforms.

  • For every fixed interior t other than 1/2, sufficiently high-frequency perturbations from any nonconstant smooth even periodic waveform give F_k(t) > F_k(1/2).The threshold can depend on t, so this is not a simultaneous assertion over all nonmidpoint weights. Irrational weight ratios leave no nonzero frequency pairs undamped, while rational ratios retain matching integer modes; fixed-order moments converge to Gaussian moments, and suitable amplitudes also yield uniform convergence of the density and its derivatives.Source: §D.2 · pp. 26–28

D.3. The scale of the dip near the midpoint.

  • For each fixed s ≠ 0 and sufficiently large k, F_k(1/2 + s/k) > F_k(1/2), with a cosine-profile width of order k^-1 and depth of order ϵ^4.The compact-interval scaling limit does not control the error relative to s^2, so it does not determine the finite-k midpoint second derivative.Source: §D.3 · p. 28

Appendix D. Periodic perturbations and a scaling limit near the midpoint.

  • The perturbation mechanism depends on the weight scale: fixed weights require matching integer frequencies, while t = 1/2 + s/k yields a continuous limiting profile.For the cosine waveform, the normalized midpoint entropy deficit tends to 1/16, whereas at each fixed interior nonmidpoint weight it tends to zero; convergence is not uniform near the endpoints.Source: §D.3 · p. 28

Appendix E. Strictly positive midpoint curvature

Appendix E constructs a Hermite family with strictly positive midpoint curvature, making equal weighting a strict local minimum. It also establishes exact moment matching and examines fixed-weight entropy comparisons.

Appendix E. Strictly positive midpoint curvature

  • A Hermite family requiring no moment correction has F''(1/2) > 0, so equal weighting is a strict local minimum.The proof estimates both the output perturbation and its second derivative with respect to the weight.Source: Appendix E · pp. 28, 31
  • For sufficiently large n, the symmetric, strongly log-concave density matches the standard Gaussian's moments through order n − 1 exactly.Hermite orthogonality makes the perturbation integrate to zero against every polynomial of degree less than n.Source: Appendix E · p. 29
  • The construction preserves symmetry, unit variance, moment matching, and uniform convergence under Gaussian smoothing; sufficiently large-degree smoothed densities are strongly log-concave.The stated convergence includes relative densities and their fixed-order derivatives.Source: Appendix E · p. 29
  • The construction also yields a fixed-weight entropy comparison above its stated parameter threshold: for sufficiently large n, Fq,n(t) > Fq,n(1/2).The supplied passage does not specify the threshold's value.Source: Appendix E · p. 31

Appendix F. An interpolation inequality for Gaussian smoothing

Appendix F determines the exact parameter range for a Gaussian interpolation inequality and clarifies what that range implies for the symmetric entropy argument. Appendix G gives a finite strongly log-concave counterexample at noise variance fraction 17/40.

Appendix F. An interpolation inequality for Gaussian smoothing

  • For 0 < q ≤ 4/7, under the stated small-norm condition, the midpoint quadratic term is negligible relative to the linear term, but this does not establish entropy concavity.The symmetric-class concavity question for 0 < q ≤ 4/7 remains open.Source: Appendix F · p. 32
  • The inequality is proved by comparing Gaussian-kernel operators and applying Cauchy–Schwarz; necessity is tested with high-degree Hermite eigenfunctions.The smoothing quadratic form is controlled by the square-root quadratic form in the asserted range.Source: Appendix F · pp. 32, 34
  • Outside the interpolation inequality’s parameter range, high-degree Hermite functions make the ratio of its sides diverge, so no perturbation-independent constant exists.The range includes its endpoint; the main-proof parameters reduce the condition to 0 < q ≤ 4/7.Source: Appendix F · pp. 32, 34, 35

Appendix G. An explicit example with noise variance fraction 17/40

  • At noise variance fraction 17/40, the appendix constructs a strongly log-concave counterexample and gives an explicit positive lower bound for the entropy gap.The finite example demonstrates a realization of the general existence result proved in the main text.Source: Appendix G · p. 35
  • The explicit density is even, smooth, strictly positive, variance one, and strongly log-concave both before and after Gaussian smoothing.Its density ratio to the standard Gaussian differs uniformly from one by less than 2.013 · 10^-1777.Source: Appendix G · p. 35
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