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On the maximal L1 influence of real-valued boolean functions

Andrew J. Young, Henry D. Pfister

TL;DR

The paper extends the KKL paradigm to real-valued Boolean functions under $p$-biased measures by establishing a sharp lower bound on the maximal $L^1$ influence of coordinates, proportional to the function variance times $\ln n / n$, with constants depending on spectral mass and smoothing parameters. The core method combines $p$-biased Fourier analysis, hypercontractivity, and a large-derivative argument (via Rossignol’s relation) to connect the rate of growth of $\mathbb{E}[f]$ with coordinate perturbations, accommodating a broad class of functions and all $0<p<1$. The authors also discuss monotone and symmetric structures, presenting weakly monotone/symmetric conditions under which their bounds hold, and illustrate the tightness of constants using Tribes functions. Overall, the work broadens KKL-type results to the $L^1$ landscape for real-valued Boolean functions and clarifies the role of spectral distribution and monotonicity in threshold phenomena.

Abstract

We show that any sequence of well-behaved (e.g. bounded and non-constant) real-valued functions of $n$ boolean variables $\{f_n\}$ admits a sequence of coordinates whose $L^1$ influence under the $p$-biased distribution, for any $p\in(0,1)$, is $Ω(\text{var}(f_n) \frac{\ln n}{n})$.

On the maximal L1 influence of real-valued boolean functions

TL;DR

The paper extends the KKL paradigm to real-valued Boolean functions under -biased measures by establishing a sharp lower bound on the maximal influence of coordinates, proportional to the function variance times , with constants depending on spectral mass and smoothing parameters. The core method combines -biased Fourier analysis, hypercontractivity, and a large-derivative argument (via Rossignol’s relation) to connect the rate of growth of with coordinate perturbations, accommodating a broad class of functions and all . The authors also discuss monotone and symmetric structures, presenting weakly monotone/symmetric conditions under which their bounds hold, and illustrate the tightness of constants using Tribes functions. Overall, the work broadens KKL-type results to the landscape for real-valued Boolean functions and clarifies the role of spectral distribution and monotonicity in threshold phenomena.

Abstract

We show that any sequence of well-behaved (e.g. bounded and non-constant) real-valued functions of boolean variables admits a sequence of coordinates whose influence under the -biased distribution, for any , is .
Paper Structure (8 sections, 6 theorems, 77 equations)

This paper contains 8 sections, 6 theorems, 77 equations.

Table of Contents

  1. Introduction
  2. Main result
  3. Fourier analysis on the $p$-biased hypercube
  4. Proof of Theorem \ref{['theorem_real_kkl']}
  5. Large derivatives
  6. Conditions implying monotonicity
  7. Weak conditions
  8. Tribes

Key Result

Theorem 1

Let $\mu$ be the $p$-biased measure, $f_{n} : \{-1,1\}^{n} \rightarrow \mathbb{R}$ and $f_{n}^{(i)} = f_{n} - E_{i}\left[ f_{n} \right]$. If $\mathrm{var}(f_{n})$ is strictly positive and $o(n^{\varepsilon})$, for all $\varepsilon > 0$, then where $\rho_{2}(\alpha) = \rho \left( e^{\alpha} + 1 \right)$ and $\rho$ is any of the smoothing parameters in Theorem theorem_p-hypercontractivity.

Theorems & Definitions (11)

  • Theorem 1
  • Theorem 2
  • proof
  • Corollary 3
  • proof
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • Lemma 6
  • ...and 1 more