Mirror Diffusion Models for Constrained and Watermarked Generation

• URL: http://arxiv.org/abs/2310.01236v2
• Date: Thu, 29 Feb 2024 06:07:33 GMT
• Title: Mirror Diffusion Models for Constrained and Watermarked Generation
• Authors: Guan-Horng Liu, Tianrong Chen, Evangelos A. Theodorou, Molei Tao
• Abstract summary: Mirror Diffusion Models (MDM) is a new class of diffusion models that generate data on convex constrained sets without losing tractability. For safety and privacy purposes, we also explore constrained sets as a new mechanism to embed invisible but quantitative information in generated data. Our work brings new algorithmic opportunities for learning tractable diffusion on complex domains.
• Score: 41.27274841596343
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Modern successes of diffusion models in learning complex, high-dimensional data distributions are attributed, in part, to their capability to construct diffusion processes with analytic transition kernels and score functions. The tractability results in a simulation-free framework with stable regression losses, from which reversed, generative processes can be learned at scale. However, when data is confined to a constrained set as opposed to a standard Euclidean space, these desirable characteristics appear to be lost based on prior attempts. In this work, we propose Mirror Diffusion Models (MDM), a new class of diffusion models that generate data on convex constrained sets without losing any tractability. This is achieved by learning diffusion processes in a dual space constructed from a mirror map, which, crucially, is a standard Euclidean space. We derive efficient computation of mirror maps for popular constrained sets, such as simplices and $\ell_2$-balls, showing significantly improved performance of MDM over existing methods. For safety and privacy purposes, we also explore constrained sets as a new mechanism to embed invisible but quantitative information (i.e., watermarks) in generated data, for which MDM serves as a compelling approach. Our work brings new algorithmic opportunities for learning tractable diffusion on complex domains. Our code is available at https://github.com/ghliu/mdm

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