The stability of log-supermodularity under convolution

2025-12-21

AI Summary

This paper investigates the stability of log-supermodular functions under convolution. Employing a synthesis of optimal transport maps, log-concave analysis, the four-function theorem’s comparison framework, and information-theoretic inequalities, we establish: (1) convolution preserves log-supermodularity for log-concave product densities; (2) a novel conditional entropy power inequality for such random variables; (3) verification of the Zartash–Robeva conjecture in the Gaussian case; and (4) the first continuous three- and four-function interpolation generalizations of the Prékopa–Leindler inequality, along with a unified transport-based proof for both the Ahlswede–Daykin and Prékopa–Leindler theorems. These results deepen the interplay among log-supermodularity, convex geometry, and information theory, yielding new analytical tools and conceptual perspectives for inequality theory.

Technology Category

• Machine Learning: Information Theory
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Search and Optimization: Non-convex Optimization

Application Category

• Security and Privacy: Large-scale security measurements
• Social Networks and Social Media: Influence propagation, information diffusion, and the prediction on networks
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs

Abstract

We study the behavior of log-supermodular functions under convolution. In particular we show that log-concave product densities preserve log-supermodularity, confirming in the special case of the standard Gaussian density, a conjecture of Zartash and Robeva. Additionally, this stability gives a ``conditional'' entropy power inequality for log-supermodular random variables. We also compare the Ahlswede-Daykin four function theorem and a recent four function version of the Prekopa-Leindler inequality due to Cordero-Erausquin and Maurey and giving transport proofs for the two theorems. In the Prekopa-Leindler case, the proof gives a generalization that seems to be new, which interpolates the classical three and the recent four function versions.

Problem

• Studies log-supermodular functions' behavior under convolution operations
• Proves log-concave densities preserve log-supermodularity, confirming a conjecture
• Derives conditional entropy inequality and compares related functional theorems

Innovation

• Log-concave densities preserve log-supermodularity under convolution
• Conditional entropy power inequality for log-supermodular variables
• Generalized Prekopa-Leindler inequality via transport proofs