bibkey: hirche2020renyibounds
authors: Christoph Hirche
year: 2020
title: "Rényi Bounds on Information Combining"
doi: 10.48550/arXiv.2004.14408
url: https://arxiv.org/abs/2004.14408v1
claim: "Conjectures IV.6 and IV.18 ask for separate convexity and concavity thresholds for binary Arimoto and Cachin Rényi information combining."
strata_touched:
  - D5/S3/Entropy/RenyiCombining/HircheCachinCombiningConcavity
  - D5/S3/Entropy/RenyiCombining/HircheArimotoCombining
license: citation-only
triage: anchor

Rényi bounds on information combining

The source is C. Hirche, “Rényi Bounds on Information Combining”, ISIT 2020, arXiv:2004.14408v1, Eq. (23); §IV-A, Eq. (47), Conjecture IV.6; and §IV-D, Conjecture IV.18, Eq. (88). This note does not assert an independent literature or priority search.

Locator

DOI: 10.48550/arXiv.2004.14408 Source: https://arxiv.org/abs/2004.14408v1 Eq. (23); §IV-A, Eq. (47), Conjecture IV.6; §IV-D, Conjecture IV.18, Eq. (88).

Arimoto information combining

Equation (23) defines $K_{\alpha }^A(X\mid Y)=e^{((1-\alpha )/\alpha )H_{\alpha }^A(X\mid Y)}$. For binary $X\sim \{p,1-p\}$, its coordinate is

$$k_{\alpha }^A(p)=(p^{\alpha }+(1-p{)}^{\alpha }{)}^{1/\alpha },p\star q=p+q-2pq.$$

The source writes $(k_{\alpha }^A{)}^{-1}$ for the inverse branch on $[0,1/2]$. Its coordinate domain is the closed interval with endpoints $1$ and ${\delta }_{\alpha }=2^{(1-\alpha )/\alpha }$; equivalently, $I_{\alpha }=[\min (1,{\delta }_{\alpha }),\max (1,{\delta }_{\alpha })]$. The implementation specifies this source inverse using Function.invFunOn. Equation (47) is the combining expression

$$k_{\alpha }^A((k_{\alpha }^A{)}^{-1}(x)\star (k_{\alpha }^A{)}^{-1}(y)).$$

Conjecture IV.6 states: “There exists a value $1<\hat{\alpha }<1.5783$, such that” this expression “is convex for $0<\alpha <1$ and concave for $1<\alpha \le \hat{\alpha }$.” The assertion is separate in each coordinate on $I_{\alpha }$. The order $\alpha =1$ is excluded.

The hyperbolic elasticity and its analytic monotonicity certificate are repository proof constructions, rather than definitions or results attributed to this source.

Cachin information combining

Conjecture IV.18 states verbatim: “There exists a value $1<\hat{\alpha }<1.3863$, such that $h_{\alpha }(h_{\alpha }^{-1}(x)\ast h_{\alpha }^{-1}(y))$ is convex for $0<\alpha <\hat{\alpha }$ and concave for $\alpha \ge 2$.”

For real $\alpha >0$, use natural logarithms and real powers. At $\alpha =1$, put $h_1(p)=\mathrm{binEntropy}(p)$, the binary Shannon entropy; at the other orders use:

$$h_{\alpha }(p)=\frac{\log (p^{\alpha }+(1-p{)}^{\alpha })}{1-\alpha },p\star q=p+q-2pq.$$

The inverse $h_{\alpha }^{-1}$ is the branch from $[0,\log 2]$ to $[0,1/2]$. For every $y\in [0,\log 2]$, the source asks for convexity below its threshold and concavity at orders at least two on $[0,\log 2]$ of $x\mapsto h_{\alpha }(h_{\alpha }^{-1}(x)\star h_{\alpha }^{-1}(y))$. Both clauses include the entropy endpoints; the concavity clause includes $\alpha =2$.

Unconditional collision entropy

The same primary text, PDF p. 2, Section III, equation (8), defines $H_{\alpha }(X)=(1-\alpha {)}^{-1}\log {\sum }_{x}p(x{)}^{\alpha }$ for an arbitrary finite alphabet. With $\alpha =2$ and base-two logarithms this is $H_{\mathrm{col}}(X)=-{\log }_2{\sum }_{x}p(x{)}^2$. For independent copies the sum is their equality probability. The conditioned entropy choices of (10)–(13) play no role in this definition.

Actual pair energy, Section 23 uses this standard entropy for a measurable address partition of the actual matrix-word law. Its packing constant, original self-spin sum and recovery-memory implication are applications derived in that volume; no general entropy definition is claimed as new.

Shannon entropy near the history length alone does not imply the same for collision entropy. Among $2^n$ equiprobable histories, merge $M_n=\lfloor 2^n/n\rfloor$ into one address and leave the others separate. Writing $p=M_n{2}^{-n}$ gives

$$H_{\mathrm{Sh}}=p{\log }_2(1/p)+(1-p)n=n-1+o(1),H_{\mathrm{col}}=-{\log }_2[p^2+(2^n-M_n)4^{-n}]=2{\log }_{2}n+o(1).$$

This elementary finite-law example concerns unrestricted partitions; it makes no assertion that the repository’s matrix-word partition has this mass pattern. Jensen’s inequality gives $H_{\mathrm{col}}\le H_{\mathrm{Sh}}$ but supplies no reverse bound at the strength needed for the critical pair estimate.