Biased Linearity Testing in the 1% Regime
Subhash Khot, Kunal Mittal
TL;DR
This work characterizes when biased linearity testing on the $p$-biased hypercube can succeed in the 1% regime by identifying a sharp bias–query threshold. It shows that for any $p\in(0,1)$ and $k>1+\frac{1}{\min\{p,1-p\}}$, there exists a distribution $\nu\in\mathcal{D}(p,k)$ with full even-weight support and pairwise independence such that the tester Lin$(\nu)$ passes with nontrivial probability on functions close to linear, forcing those functions to correlate with a linear form under $\mu_p^{\otimes n}$. Conversely, if $k<1+\frac{1}{\min\{p,1-p\}}$, no such test exists. The paper develops a Gaussian variant to construct counterexamples, proves a key necessity of pairwise independence, and lays out a robust analytic framework (involving Hermite polynomials, invariance principles, and list decoding) to analyze the Lin$(\nu)$ test and its limitations. Together, these results reveal a fundamental tradeoff between query complexity and the bias parameter in tolerant linearity testing and extend the understanding of testing under nonuniform input distributions.
Abstract
We study linearity testing over the $p$-biased hypercube $(\{0,1\}^n, μ_p^{\otimes n})$ in the 1% regime. For a distribution $ν$ supported over $\{x\in \{0,1\}^k:\sum_{i=1}^k x_i=0 \text{ (mod 2)} \}$, with marginal distribution $μ_p$ in each coordinate, the corresponding $k$-query linearity test $\text{Lin}(ν)$ proceeds as follows: Given query access to a function $f:\{0,1\}^n\to \{-1,1\}$, sample $(x_1,\dots,x_k)\sim ν^{\otimes n}$, query $f$ on $x_1,\dots,x_k$, and accept if and only if $\prod_{i\in [k]}f(x_i)=1$. Building on the work of Bhangale, Khot, and Minzer (STOC '23), we show, for $0 < p \leq \frac{1}{2}$, that if $k \geq 1 + \frac{1}{p}$, then there exists a distribution $ν$ such that the test $\text{Lin}(ν)$ works in the 1% regime; that is, any function $f:\{0,1\}^n\to \{-1,1\}$ passing the test $\text{Lin}(ν)$ with probability $\geq \frac{1}{2}+ε$, for some constant $ε> 0$, satisfies $\Pr_{x\sim μ_p^{\otimes n}}[f(x)=g(x)] \geq \frac{1}{2}+δ$, for some linear function $g$, and a constant $δ= δ(ε)>0$. Conversely, we show that if $k < 1+\frac{1}{p}$, then no such test $\text{Lin}(ν)$ works in the 1% regime. Our key observation is that the linearity test $\text{Lin}(ν)$ works if and only if the distribution $ν$ satisfies a certain pairwise independence property.