The stability of log-supermodularity under convolution
Mokshay Madiman, James Melbourne, Cyril Roberto
TL;DR
The paper investigates how log-supermodularity behaves under convolution, proving that convolving with a log-concave product density preserves log-supermodularity (including the Gaussian case and discrete lattice versions). It links this stability to a conditional entropy power inequality for dependent variables and develops transport-based proofs for four-function inequalities, drawing connections to non-linear extensions of Prekopa-Leindler. The results unify log-supermodularity with information-theoretic inequalities across continuous and discrete settings and introduce interpolation between three- and four-function Prekopa-Leindler-type statements. These insights enhance understanding of how submodular-like properties propagate under convolution and transport.
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.