Noise sensitivity from fractional query algorithms and the axis-aligned Laplacian

TL;DR

This paper introduces classical fractional query algorithms, linking their complexity and noise sensitivity to PDEs, and demonstrating potential advantages over traditional decision trees in average-case scenarios.

Contribution

It generalizes decision trees through fractional query algorithms, establishes their complexity via PDEs, and connects noise sensitivity with Fourier analysis and PDEs.

Findings

Fractional query algorithms can outperform decision trees.

Complexity characterized by a nonlinear PDE.

Noise sensitivity linked to Fourier weights and PDEs.

Abstract

We introduce the notion of classical fractional query algorithms, which generalize decision trees in the average-case setting, and can potentially perform better than them. We show that the limiting run-time complexity of a natural class of these algorithms obeys the non-linear partial differential equation $\min_{k}\partial^{2}u/\partial x_{k}^{2}=-2$, and that the individual bit revealment satisfies the Schramm-Steif bound for Fourier weight, connecting noise sensitivity with PDEs. We discuss relations with other decision tree results.