The relative-error property testing model was introduced in [CDHLNSY24] to facilitate the study of property testing for "sparse" Boolean-valued functions, i.e. ones for which only a small fraction of all input assignments satisfy the function. In this framework, the distance from the unknown target function $f$ that is being tested to a function $g$ is defined as $\mathrm{Vol}(f \mathop{\triangle} g)/\mathrm{Vol}(f)$, where the numerator is the fraction of inputs on which $f$ and $g$ disagree and the denominator is the fraction of inputs that satisfy $f$. Recent work [CDHNSY26] has shown that over the Boolean domain $\{0,1\}^n$, any relative-error testing algorithm for the fundamental class of halfspaces (i.e. linear threshold functions) must make $Ω(\log n)$ oracle calls. In this paper we complement the [CDHNSY26] lower bound by showing that halfspaces can be relative-error tested over $\mathbb{R}^n$ under the standard $N(0,I_n)$ Gaussian distribution using a sublinear number of oracle calls -- in particular, substantially fewer than would be required for learning. Our results use a wide range of tools including Hermite analysis, Gaussian isoperimetric inequalities, and geometric results on noise sensitivity and surface area.

翻译：相对误差属性测试模型由[CDHLNSY24]提出，旨在研究“稀疏”布尔值函数（即仅少量输入赋值满足该函数的函数）的属性测试。在该框架下，被测试的未知目标函数$f$与函数$g$之间的距离定义为$\mathrm{Vol}(f \mathop{\triangle} g)/\mathrm{Vol}(f)$，其中分子是$f$和$g$取值不同的输入所占比例，分母是满足$f$的输入所占比例。近期工作[CDHNSY26]表明，在布尔域$\{0,1\}^n$上，对于半空间（即线性阈值函数）这一基础类别的任何相对误差测试算法必须进行$\Omega(\log n)$次预言机调用。本文通过证明在标准$N(0,I_n)$高斯分布下，半空间可在$\mathbb{R}^n$上使用次线性次数的预言机调用进行相对误差测试，从而补充了[CDHNSY26]的下界结果——特别是，所需调用次数远少于学习所需的次数。我们的结果使用了广泛工具，包括厄米分析、高斯等周不等式以及噪声敏感性和表面积方面的几何结论。