Curvature methods reveal new inequalities for weakly dependent Boolean measures
Discrete Isoperimetric Inequalities via Curvature
Discrete Mathematics
Summary
Isoperimetric inequalities help understand how groups of yes/no decisions behave when you change just a few of them. These inequalities were known mostly for simple cases where each decision is independent. The authors developed a new approach using curvature ideas that works even when decisions weakly influence each other. Their work covers certain statistical physics models and can apply to more general situations. This opens the door to better analyzing complex systems with dependencies.
What this means in practice
- •For statistical physics modelers: Analyze dependency structures in Ising models within the uniqueness regime to better understand phase behavior.
- •For machine learning researchers: Develop tighter bounds on generalization or robustness in models with weakly dependent binary features using new inequalities.
A theory result. No direct application yet.
Authors
Zejia Chen, Zongchen Chen, Xinyuan Zhang
Abstract
Isoperimetric inequalities on the Boolean hypercube play a fundamental role in the analysis of Boolean functions. These inequalities have been established primarily for the uniform measure and for biased product measures, often through Fourier-analytic or inductive arguments. We develop a curvature-based semigroup framework to establish isoperimetric inequalities for measures with weakly dependent coordinates. In particular, we show that a Dobrushin-type condition, together with marginal boundedness of the distribution, implies the local Bobkov inequality, Talagrand's $L^1$--$L^2$ and variance--surface-area inequalities, the Kahn--Kalai--Linial inequality, and the Eldan--Gross inequality. Our results apply to zero-field Ising models with interaction matrix $J$ throughout the Dobrushin uniqueness regime $\|J\|_1<1$. The framework builds on discrete Bakry--Émery theory and gradient estimates and can also be extended to measures on Hamming slices or hypergrids.