Mirror Diffusion Models for Constrained and Watermarked Generation
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866929258920673280 |
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| author | Liu, Guan-Horng Chen, Tianrong Theodorou, Evangelos A. Tao, Molei |
| author_facet | Liu, Guan-Horng Chen, Tianrong Theodorou, Evangelos A. Tao, Molei |
| contents | 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 |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_01236 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Mirror Diffusion Models for Constrained and Watermarked Generation Liu, Guan-Horng Chen, Tianrong Theodorou, Evangelos A. Tao, Molei Machine Learning Computer Vision and Pattern Recognition 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 |
| title | Mirror Diffusion Models for Constrained and Watermarked Generation |
| topic | Machine Learning Computer Vision and Pattern Recognition |
| url | https://arxiv.org/abs/2310.01236 |