de Almeida-Thouless Stability and Replica Symmetry Breaking in a Hybrid Watermarking Model with Image and Message Decoding

Abstract

We investigate the de Almeida-Thouless (AT) stability of the replica symmetric (RS) solution for a spread-spectrum watermarking model incorporating Bayes-estimation-based image restoration. In our previously proposed model [1], which simultaneously performs message decoding and image restoration under Gaussian channel noise, a discrepancy between the theoretical predictions of the RS solution and Markov chain Monte Carlo (MCMC) simulations was observed in specific parameter regimes. By analyzing the blind case where both the original message and image are unknown, we explicitly derive the conditions for the AT stability of the RS solution by clarifying the structure of the replicon eigenspace for the hybrid system. Our results reveal that AT stability is broken precisely in the parameter region where the RS solution deviates from MCMC simulations, thereby successfully explaining the mechanism underlying this discrepancy in terms of replica symmetry breaking (RSB). [1] M. Kawamura, T. Uezu, K. Hayashi, and M. Okada, ``Statistical mechanical evaluation of a spread-spectrum watermarking model with image restoration,'' Phys. Rev. E \textbf{99}, 062132 (2019).