On the sum of $L1$ influences

TL;DR

This paper proves that the total $L_1$ influence of bounded functions over the discrete cube can be bounded polynomially in their degree, resolving an open problem and introducing a new analytic approach.

Contribution

The paper introduces a novel quantity $ ext{I}_p(f)$ to bound $L_1$ influences, solving an open problem by Aaronson and Ambainis.

Findings

Total $L_1$ influence is polynomially bounded by degree for bounded functions.

Introduces a new analytic quantity $ ext{I}_p(f)$ to analyze influences.

Application to graph theory and connections to quantum query complexity.

Abstract

For a function $f$ over the discrete cube, the total $L_1$ influence of $f$ is defined as $\sum_{i=1}^n \|\partial_i f\|_1$, where $\partial_i f$ denotes the discrete derivative of $f$ in the direction $i$. In this work, we show that the total $L_1$ influence of a $[-1,1]$-valued function $f$ can be upper bounded by a polynomial in the degree of $f$, resolving affirmatively an open problem of Aaronson and Ambainis (ITCS 2011). The main challenge here is that the $L_1$ influences do not admit an easy Fourier analytic representation. In our proof, we overcome this problem by introducing a new analytic quantity $\mathcal I_p(f)$, relating this new quantity to the total $L_1$ influence of $f$. This new quantity, which roughly corresponds to an average of the total $L_1$ influences of some ensemble of functions related to $f$, has the benefit of being much easier to analyze, allowing us to resolve the problem of Aaronson and Ambainis. We also give an application of the theorem to graph theory, and discuss the connection between the study of bounded functions over the cube and the quantum query complexity of partial functions where Aaronson and Ambainis encountered this question.