In this short note, we initiate the study of the Linear Isomorphism Testing Problem in the setting of communication complexity, a natural linear algebraic generalization of the classical Equality problem. Given Boolean functions $f, g : \mathbb{F}_2^n \to \{-1, +1\}$, Alice and Bob are tasked with determining whether $f$ and $g$ are equivalent up to a nonsingular linear transformation of the input variables, or far from being so. This problem has been extensively investigated in several models of computation, including standard algorithmic and property testing frameworks, owing to its fundamental connections with combinatorial circuit design, complexity theory, and cryptography. However, despite its broad relevance, it has remained unexplored in the context of communication complexity, a gap we address in this work. Our main results demonstrate that the approximate spectral norm of the input functions plays a central role in governing the communication complexity of this problem. We design a simple deterministic protocol whose communication cost is polynomial in the approximate spectral norm, and complement it with nearly matching lower bounds (up to a quadratic gap). In the randomised setting with private coins, we present an even more efficient protocol, though equally simple, that achieves a quadratically improved dependence on the approximate spectral norm compared to the deterministic case, and we prove that such a dependence is essentially unavoidable. These results identify the approximate spectral norm as a key complexity measure for testing linear invariance in the communication complexity framework. As a core technical ingredient, we establish new junta theorems for Boolean functions with small approximate spectral norm, which may be of independent interest in Fourier analysis and learning theory.

翻译：在这篇短文中，我们首次在通信复杂度框架下研究线性同构检验问题——这是经典等式问题在代数上的自然推广。给定布尔函数 $f, g : \mathbb{F}_2^n \to \{-1, +1\}$，Alice 和 Bob 需要判定 $f$ 和 $g$ 是否在输入变量的非奇异线性变换下等价，抑或相距甚远。由于其与组合电路设计、复杂性理论和密码学的基础联系，该问题已在多种计算模型（包括标准算法框架和性质检验框架）中得到广泛研究。然而，尽管具有广泛的相关性，该问题在通信复杂度背景下仍未被探索，这正是本文所要填补的空白。我们的主要结果表明，输入函数的近似谱范数在决定该问题的通信复杂度中起着核心作用。我们设计了一个简单的确定性协议，其通信代价与近似谱范数呈多项式关系，并辅以近乎匹配的下界（至多存在二次差距）。在具有私有随机性的随机化设置中，我们提出了一个更为高效且同样简单的协议，相比确定性情形，其对近似谱范数的依赖实现了二次改进，并证明这种依赖本质上是不可避免的。这些结果确立了近似谱范数作为通信复杂度框架下检验线性不变性的关键复杂性度量。作为核心技术要素，我们为具有小近似谱范数的布尔函数建立了新的 Junta 定理，该结果在傅里叶分析与学习理论中可能具有独立价值。