Fundamental Trade-Offs in Multi-Bit Watermarking of Stochastic Processes

2026-05-09

AI Summary

This work addresses the fundamental trade-offs among low false-positive rates, high-quality generation (preserving output distribution fidelity), and reliable decoding in multi-bit watermarking for high-stakes applications. The authors formulate watermark embedding as a problem of embedding information into distributions and cast detection as a multi-hypothesis testing task under distortion and rate constraints, establishing a unified information-theoretic framework that characterizes the intrinsic interplay among false-positive probability, detection error rate, distortion, and embedded information rate. They derive, for the first time, a model-agnostic theoretical benchmark applicable to any watermarking scheme, providing finite-sample achievability and converse theorems along with asymptotic limits for stationary ergodic processes. Experimental validation using a reference scheme satisfying the theoretical assumptions confirms the predicted fundamental trade-offs.

Abstract

We study multi-bit watermarking for data generated by stochastic processes, where a hidden message is embedded during sampling and must be decodable by an authorized detector that possesses side information unavailable to unauthorized observers. In high-stakes deployments, a practical watermark must simultaneously control false alarms, preserve generation quality without distorting the output distribution, and support reliable multi-bit decoding. Satisfying all three goals at once inevitably creates fundamental trade-offs. We formulate watermark embedding as a distributional information-embedding problem and watermark detection as a multiple-hypothesis testing problem under distortion and rate constraints, leading to four fundamental metrics: false-alarm probability, detection error probability, distortion, and information rate. Within this information-theoretic framework, we derive matched converse and achievability bounds that characterize the optimal trade-offs and provide scheme-agnostic benchmarks for any watermarking method. For stationary ergodic stochastic processes, we further obtain matched asymptotic limits and connect them to the finite-sample regime. Finally, we present a reference watermarking construction satisfying our assumptions and empirically illustrating the predicted trade-offs.

Problem

• multi-bit watermarking
• stochastic processes
• false-alarm probability
• distortion
• information rate

Innovation

• multi-bit watermarking
• stochastic processes
• information-theoretic trade-offs
• distributional embedding
• multiple-hypothesis testing