Combinatorial Entropy Power Inequalities: A Preliminary Study of the Stam region

Abstract

We initiate the study of the Stam region, defined as the subset of the positive orthant in ${\mathbb{R}}^{2^n-1}$ that arises from considering entropy powers of subset sums of $n$ independent random vectors in a Euclidean space of finite dimension. We show that the class of fractionally superadditive set functions provides an outer bound to the Stam region, resolving a conjecture of A. R. Barron and the first author. On the other hand, the entropy power of a sum of independent random vectors is not supermodular in any dimension. We also develop some qualitative properties of the Stam region, showing for instance that its closure is a logarithmically convex cone.

arXiv comment16 pages

References in corpus (24)

• Generalized Entropy Power Inequalities and Monotonicity Properties of Information
• Information Inequalities for Joint Distributions, with Interpretations and Applications
• Fisher Information inequalities and the Central Limit Theorem
• Reverse Brunn-Minkowski and reverse entropy power inequalities for convex measures
• Beyond the entropy power inequality, via rearrangements
• Forward and Reverse Entropy Power Inequalities in Convex Geometry
• Sumset and Inverse Sumset Inequalities for Differential Entropy and Mutual Information
• Log-concavity, ultra-log-concavity, and a maximum entropy property of discrete compound Poisson measures
• Compound Poisson Approximation via Information Functionals
• Entropy bounds on abelian groups and the Ruzsa divergence
• On the problem of reversibility of the entropy power inequality
• Dimensional behaviour of entropy and information
• Entropy and set cardinality inequalities for partition-determined functions
• Cores of Cooperative Games in Information Theory
• Rogozin's convolution inequality for locally compact groups
• Monotonic Convergence in an Information-Theoretic Law of Small Numbers
• Entropy Inequalities for Sums in Prime Cyclic Groups
• Majorization and Rényi Entropy Inequalities via Sperner Theory
• Do Minkowski averages get progressively more convex?
• Entropy power inequality for a family of discrete random variables
• An algorithmic and a geometric characterization of Coarsening At Random
• An Algorithmic and a geometric characterization of coarsening at random
• Thinning, photonic beamsplitting, and a general discrete entropy power inequality
• The convexification effect of Minkowski summation

Cited by in corpus (13)

• Forward and Reverse Entropy Power Inequalities in Convex Geometry
• Entropy bounds on abelian groups and the Ruzsa divergence
• Entropy Inequalities for Sums in Prime Cyclic Groups
• Majorization and Rényi Entropy Inequalities via Sperner Theory
• The Differential Entropy of Mixtures: New Bounds and Applications
• Sharp moment-entropy inequalities and capacity bounds for log-concave distributions
• Two remarks on generalized entropy power inequalities
• On the volume of the Minkowski sum of zonoids
• Volumes of subset Minkowski sums and the Lyusternik region
• Conditional Rényi entropy and the relationships between Rényi capacities
• An entropy inequality for symmetric random variables
• Sumset estimates in convex geometry
• The norm of the Fourier transform on compact or discrete abelian groups