Optimal Watermark Generation under Type I and Type II Errors

TL;DR

This paper formulates watermarking as a hypothesis testing problem, deriving bounds on fidelity loss and optimal watermarking strategies under Type I and II error constraints, providing a theoretical foundation for watermark design.

Contribution

It introduces a theoretical framework for watermarking based on statistical hypothesis testing, establishing bounds and optimal methods for minimal fidelity distortion.

Findings

Derived a tight lower bound on fidelity loss under error constraints

Characterized the optimal watermarked distribution achieving this bound

Developed an optimal sampling rule for watermark insertion

Abstract

Watermarking has recently emerged as a crucial tool for protecting the intellectual property of generative models and for distinguishing AI-generated content from human-generated data. Despite its practical success, most existing watermarking schemes are empirically driven and lack a theoretical understanding of the fundamental trade-off between detection power and generation fidelity. To address this gap, we formulate watermarking as a statistical hypothesis testing problem between a null distribution and its watermarked counterpart. Under explicit constraints on false-positive and false-negative rates, we derive a tight lower bound on the achievable fidelity loss, measured by a general $f$-divergence, and characterize the optimal watermarked distribution that attains this bound. We further develop a corresponding sampling rule that provides an optimal mechanism for inserting watermarks with minimal fidelity distortion. Our result establishes a simple yet broadly applicable principle linking hypothesis testing, information divergence, and watermark generation.