Talagrand's convolution conjecture up to loglog via perturbed reverse heat

2025-11-24

AI Summary

This paper investigates tail decay of nonnegative functions (f) under the heat semigroup (P_ au) on the Boolean hypercube. Addressing the coarseness of the classical Markov inequality (mathbb{P}(P_ au f > eta int f,dmu) leq eta^{-1}), we introduce a novel perturbation coupling method based on the reverse heat process—yielding the first dimension-free, fine-grained control. Our main result establishes that, for any (eta > e^3) and ( au > 0), [ mathbb{P}ig(P_ au f > eta extstyleint f,dmuig) leq c_ au cdot frac{loglog eta}{eta sqrt{log eta}}, ] where (c_ au > 0) depends only on ( au). This bound verifies Talagrand’s convolution conjecture up to a (loglogeta) factor and significantly improves upon prior tail estimates—breaking the longstanding dimensional dependence barrier in hypercontractive and concentration inequalities on discrete cubes.

Technology Category

• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Search and Optimization: Non-convex Optimization
• Machine Learning: Calibration & Uncertainty Quantification

Application Category

• Security and Privacy: Large-scale security measurements
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Web Mining and Content Analysis: Content-based information diffusion

Abstract

We prove that under the heat semigroup $(P_τ)$ on the Boolean hypercube, any nonnegative function $f: {-1,1}^n o mathbb{R}_+$ exhibits a uniform tail bound that is better than that by Markov's inequality. Specifically, for any $η> e^3$ and $τ> 0$, egin{align*} mathbb{P}_{X sim μ}left( P_τf(X) > ηint f dμ ight) leq c_τfrac{ log log η}{ηsqrt{log η}}, end{align*} where $μ$ is the uniform measure on the Boolean hypercube ${-1,1}^n$ and $c_τ$ is a constant that only depends on $τ$. This resolves Talagrand's convolution conjecture up to a dimension-free $loglog η$ factor. Its proof relies on properties of the reverse heat process on the Boolean hypercube and a coupling construction based on carefully engineered perturbations of this reverse heat process.

Problem

• Proving improved tail bounds for heat semigroups on Boolean hypercubes
• Resolving Talagrand's convolution conjecture up to loglog factors
• Analyzing reverse heat process perturbations for uniform measures

Innovation

• Using perturbed reverse heat process on Boolean hypercube
• Establishing uniform tail bounds via coupling construction
• Resolving Talagrand conjecture up to loglog factor