Talagrand-Type Correlation Inequalities for Supermodular and Submodular Functions on the Hypercube
Fan Chang, Yu Chen
TL;DR
The paper establishes Talagrand-type correlation inequalities for pairs of increasing Boolean functions on the hypercube under submodularity or supermodularity, proving Cov$(f,g) \ge \frac{1}{4}\sum_i \mathrm{Inf}_i[f]\mathrm{Inf}_i[g]$ in this structured regime. It delivers two distinct proofs: a heat-semigroup representation leveraging second-order discrete derivatives to isolate Level-≥2 Fourier weight and a purely inductive argument on dimension that avoids semigroup methods, with the former extending to real-valued functions. The work connects to the Friedgut–Kahn–Kalai–Keller antipodal framework and Chvátal-type conjectures, clarifying how submodularity/supermodularity yields strengthening and enabling a fuller understanding of when higher-level Fourier weights must contribute positively. In addition, the paper provides sharp upper bounds of Talagrand type for higher-level contributions and a two-functional Poincaré inequality with Level-1 refinements, offering a comprehensive view of both lower and upper bounds for correlation in this discrete setting. These results advance the quantitative theory of correlations for structured Boolean functions and have potential implications for learning submodular functions and related combinatorial probability questions.
Abstract
Talagrand initiated a quantitative program by lower-bounding the correlation of any two increasing Boolean functions in terms of their influences, thereby capturing how strongly the functions depend on the exact coordinates. We strengthen this line of results by proving Talagrand-type correlation lower bounds that hold whenever the increasing functions additionally satisfy super/submodularity. In particular, under super/submodularity, we establish the ``dream inequality'' $$\mathbb{E}[fg]-\mathbb{E}[f]\mathbb{E}[g]\ge \frac{1}{4}\cdot\sum\limits_{i=1}^n\mathrm{Inf}_i[f]\mathrm{Inf}_i[g].$$ Thereby confirming a conjectural direction suggested by Kalai--Keller--Mossel. Our results also clarify the connection to the antipodal strengthening considered by Friedgut, Kahn, Kalai, and Keller, who showed that a famous Chvátal's conjecture is equivalent to a certain reinforcement of Talagrand-type correlation inequality when one function is antipodal. Thus, our inequality verifies the Friedgut--Kahn--Kalai--Keller conjectural bound in this structured regime (super/submodular). Our approach uses two complementary methods: (1) a semigroup proof based on a new heat-semigroup representation via second-order discrete derivatives, and (2) an induction proof that avoids semigroup argument entirely.