Let $X,X'$ be independent and identically distributed discrete real-valued random variables of finite Shannon entropy, and write $H(X)$ for the Shannon entropy of $X$. We prove that \[ \max\{H(X+X'),\,H(XX')\} \ge \frac87 H(X)-O(\log H(X)). \] This is the entropic analog of the celebrated sum-product phenomenon, and answers a question of Goh, which simply asked for a coefficient strictly larger than 1. An example by the author, Gavalakis, and Kontoyiannis showed the coefficient cannot exceed $\frac43$. Previous work by Gavalakis, Goh, and Kontoyiannis was able to prove a result of a weaker form, which could not translate to a coefficient strictly larger than 1 because of examples where the min-entropy is significantly smaller than the Shannon entropy. By splitting the distribution of $X$ into uniform pieces, which costs $O(\log H(X))$ entropy, we obviate this issue, establishing a coefficient of $\frac{10}{9}$. We augment this to $\frac87$ by adapting the work of Solymosi, which established the combinatorial sum-product phenomenon with coefficient $\frac43$ by bounding the multiplicative energy, to the entropy setting, again via a uniformization technique.
Let $X_1,X_2,\ldots$ be i.i.d. finitely supported random variables in a torsion-free abelian group, and write $S_k=X_1+\cdots+X_k$, and $H(S_k)$ is the Shannon entropy $S_k$, for all $k \ge 1$. We prove that, for every fixed $n\geq1$, \[ H(S_{n+1})-H(S_n) \geq \frac12\log\frac{n+1}{n} -o_{H(X_1)\to\infty}(1), \] unifor...
Sharp high-entropy lower bounds for the entropy of a sum were known for identically distributed summands in torsion-free abelian groups and in prime cyclic groups. For arbitrary independent summands, Gavalakis, Goh and Kontoyiannis obtained an additive constant of $1/8$ and conjectured that the sharp constant is $1/2$....
Suppose $X,Y$ are independent random variables with values in a compact abelian group $(G,+)$. We examine the following two entropy power-type inequalities: $h(X+Y)\geq \frac{1}{2}h(X)+\frac{1}{2}h(Y)$ and $h(X+Y)\geq \max\{h(X),h(Y)\}$, where the entropy $h(Z)$ of a $G$-valued random variable $Z$ is defined in terms o...
Lampros Gavalakis, I. Kontoyiannis, Sharang M. Sriramu et al.· 1 citation
It is shown that Shannon entropy, Fisher information, and quantum-mechanical kinetic energy may all be viewed as measures of correlation. We begin with a fundamental correlation-destroying map: a joint probability density $\rho(x,y)$ is replaced by $\rho_x(x)\rho_y(y)$, the product of its marginal distributions. In the...
The present work builds on the differential-equation method, itself a limiting form of the auxiliary-receiver approach in network information theory using a continuum of degraded receivers, and gives a computer-assisted proof of the Courtade--Kumar conjecture.
Zi-Jie Chen, Amin Gohari, Adel Javanmard et al.· 1 citation
Entropy functionals and their associated divergences underlie many statistical methods, including maximum entropy inference, minimum divergence estimation, and goodness-of-fit testing, yet choosing among Shannon, R\'enyi, Tsallis, and more general entropies is often a matter of convention rather than structural princip...
D. Lazarev· 0 citations
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