# Boolean Functions with Biased Inputs: Approximation and Noise Sensitivity

## TL;DR

This paper analyzes how well Boolean functions can be approximated by simpler classes like juntas and linear functions under biased input distributions, linking approximation quality to Fourier analysis and noise sensitivity.

## Contribution

It characterizes optimal approximations and mismatch probabilities for biased inputs using biased Fourier expansion, and connects these to noise sensitivity analysis.

## Key findings

- Optimal approximation strategies for biased inputs are derived.
- Mismatch probabilities are expressed via biased Fourier coefficients.
- Noise sensitivity is characterized in terms of Fourier analysis.

## Abstract

This paper considers the problem of approximating a Boolean function $f$ using another Boolean function from a specified class. Two classes of approximating functions are considered: $k$-juntas, and linear Boolean functions. The $n$ input bits of the function are assumed to be independently drawn from a distribution that may be biased. The quality of approximation is measured by the mismatch probability between $f$ and the approximating function $g$. For each class, the optimal approximation and the associated mismatch probability is characterized in terms of the biased Fourier expansion of $f$. The technique used to analyze the mismatch probability also yields an expression for the noise sensitivity of $f$ in terms of the biased Fourier coefficients, under a general i.i.d. input perturbation model.