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| Main Authors: | , |
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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2509.20511 |
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| _version_ | 1866918147460694016 |
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| author | Leong, Oscar Traonmilin, Yann |
| author_facet | Leong, Oscar Traonmilin, Yann |
| contents | Recovering high-dimensional signals from corrupted measurements is a central challenge in inverse problems. Recent advances in generative diffusion models have shown remarkable empirical success in providing strong data-driven priors, but rigorous recovery guarantees remain limited. In this work, we develop a theoretical framework for analyzing deterministic diffusion-based algorithms for inverse problems, focusing on a deterministic version of the algorithm proposed by Kadkhodaie \& Simoncelli \cite{kadkhodaie2021stochastic}. First, we show that when the underlying data distribution concentrates on a low-dimensional model set, the associated noise-convolved scores can be interpreted as time-varying projections onto such a set. This leads to interpreting previous algorithms using diffusion priors for inverse problems as generalized projected gradient descent methods with varying projections. When the sensing matrix satisfies a restricted isometry property over the model set, we can derive quantitative convergence rates that depend explicitly on the noise schedule. We apply our framework to two instructive data distributions: uniform distributions over low-dimensional compact, convex sets and low-rank Gaussian mixture models. In the latter setting, we can establish global convergence guarantees despite the nonconvexity of the underlying model set. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_20511 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Recovery Theory for Diffusion Priors: Deterministic Analysis of the Implicit Prior Algorithm Leong, Oscar Traonmilin, Yann Machine Learning Signal Processing Optimization and Control Recovering high-dimensional signals from corrupted measurements is a central challenge in inverse problems. Recent advances in generative diffusion models have shown remarkable empirical success in providing strong data-driven priors, but rigorous recovery guarantees remain limited. In this work, we develop a theoretical framework for analyzing deterministic diffusion-based algorithms for inverse problems, focusing on a deterministic version of the algorithm proposed by Kadkhodaie \& Simoncelli \cite{kadkhodaie2021stochastic}. First, we show that when the underlying data distribution concentrates on a low-dimensional model set, the associated noise-convolved scores can be interpreted as time-varying projections onto such a set. This leads to interpreting previous algorithms using diffusion priors for inverse problems as generalized projected gradient descent methods with varying projections. When the sensing matrix satisfies a restricted isometry property over the model set, we can derive quantitative convergence rates that depend explicitly on the noise schedule. We apply our framework to two instructive data distributions: uniform distributions over low-dimensional compact, convex sets and low-rank Gaussian mixture models. In the latter setting, we can establish global convergence guarantees despite the nonconvexity of the underlying model set. |
| title | A Recovery Theory for Diffusion Priors: Deterministic Analysis of the Implicit Prior Algorithm |
| topic | Machine Learning Signal Processing Optimization and Control |
| url | https://arxiv.org/abs/2509.20511 |