Diffusion models have rapidly become a vital part of deep generative architectures, given today's increasing demands. Obtaining large, high-performance diffusion models demands significant resources, highlighting their importance as intellectual property worth protecting. However, existing watermarking techniques for ownership verification are insufficient when applied to diffusion models. Very recent research in watermarking diffusion models either exposes watermarks during task generation, which harms the imperceptibility, or is developed for conditional diffusion models that require prompts to trigger the watermark. This paper introduces WDM, a novel watermarking solution for diffusion models without imprinting the watermark during task generation. It involves training a model to concurrently learn a Watermark Diffusion Process (WDP) for embedding watermarks alongside the standard diffusion process for task generation. We provide a detailed theoretical analysis of WDP training and sampling, relating it to a shifted Gaussian diffusion process via the same reverse noise. Extensive experiments are conducted to validate the effectiveness and robustness of our approach in various trigger and watermark data configurations.

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