Spectral Shadows: When Communication Complexity Meets Linear Invariance Testing

2026-01-11

arXiv.org

AI Summary

This work investigates the communication complexity of testing linear isomorphism between Boolean functions. Introducing linear isomorphism testing to the communication complexity setting for the first time, it proposes the approximate spectral norm as a central complexity measure. By combining Fourier-analytic techniques with properties of the approximate spectral norm, the authors design a deterministic protocol whose communication cost is polynomial in this norm, as well as a private-coin randomized protocol that achieves a quadratically better bound. Both protocols are shown to be nearly optimal. Furthermore, the paper establishes a new junta theorem for Boolean functions with small approximate spectral norm, revealing a deep structural connection between this norm and the function’s dependence on a limited number of variables.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Machine Learning: Kernel Methods
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Responsible Web: Measurement, analysis, and circumvention of Web censorship

Abstract

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.

Problem

• Linear Isomorphism Testing
• Communication Complexity
• Boolean Functions
• Linear Invariance
• Approximate Spectral Norm

Innovation

• communication complexity
• linear isomorphism testing
• approximate spectral norm
• junta theorems
• randomized protocols