Let $f$ and $g$ be Boolean functions over a finite Abelian group $\mathcal{G}$, where $g$ is fully known, and we have {\em query access} to $f$, that is, given any $x \in \mathcal{G}$ we can get the value $f(x)$. We study the tolerant isomorphism testing problem: given $\epsilon \geq 0$ and $\tau > 0$, we seek to determine, with minimal queries, whether there exists an automorphism $\sigma$ of $\mathcal{G}$ such that the fractional Hamming distance between $f \circ \sigma$ and $g$ is at most $\epsilon$, or whether for all automorphisms $\sigma$, the distance is at least $\epsilon + \tau$. We design an efficient tolerant testing algorithm for this problem, with query complexity $\mathrm{poly}\left( s, 1/\tau \right)$, where $s$ bounds the spectral norm of $g$. Additionally, we present an improved algorithm when $g$ is Fourier sparse. Our approach uses key concepts from Abelian group theory and Fourier analysis, including the annihilator of a subgroup, Pontryagin duality, and a pseudo inner-product for finite Abelian groups. We believe these techniques will find further applications in property testing.

翻译：设$f$和$g$为定义在有限阿贝尔群$\mathcal{G}$上的布尔函数，其中$g$完全已知，而对$f$我们仅拥有{\em查询访问权限}，即给定任意$x \in \mathcal{G}$可获取$f(x)$的值。我们研究容错同构测试问题：给定$\epsilon \geq 0$与$\tau > 0$，我们旨在以最小查询次数判定是否存在$\mathcal{G}$的自同构$\sigma$，使得$f \circ \sigma$与$g$之间的分数汉明距离至多为$\epsilon$；或者对于所有自同构$\sigma$，该距离至少为$\epsilon + \tau$。我们为此问题设计了一种高效的容错测试算法，其查询复杂度为$\mathrm{poly}\left( s, 1/\tau \right)$，其中$s$为$g$的谱范数上界。此外，当$g$具有傅里叶稀疏性时，我们提出了一种改进算法。我们的方法运用了阿贝尔群论与傅里叶分析中的核心概念，包括子群的零化子、庞特里亚金对偶性，以及有限阿贝尔群的伪内积运算。我们相信这些技术将在属性测试领域获得更广泛的应用。