将生成模型用于无限维信号压缩感知，实现与分辨率无关的稳定重建。
Infinite dimensional generative sensing
- 在希尔伯特空间中定义无限维局部相干性，推导出最优采样分布
- 测量数仅需与先验内在维度相关，与环境维度无关（对数因子内）
- 低分辨率生成器可作隐式正则化，提升严重欠采样下的重建稳定性
深度生成模型已成为反问题中建模先验的标准方法，超越了传统的稀疏性方法。然而，现有理论保证大多局限于有限维向量空间，当物理信号被建模为希尔伯特空间中的函数时，这一差距便显现。本文提出一个严谨的希尔伯特空间中生成压缩感知框架。我们将在有限维中定义的局部相干性推广至无限维情形，推导出最优且与分辨率无关的采样分布。借助广义受限等距性质，证明当测量数与先验内在维度成比例（含对数因子）时，可实现稳定恢复，且独立于环境维度。最后，基于达西流方程的数值实验验证了理论结果，并表明在严重欠采样条件下，使用低分辨率生成器可作为隐式正则化，提升重建稳定性。
原文摘要 · Abstract (English)
Deep generative models have become a standard for modeling priors for inverse problems, going beyond classical sparsity-based methods. However, existing theoretical guarantees are mostly confined to finite-dimensional vector spaces, creating a gap when the physical signals are modeled as functions in Hilbert spaces. This work presents a rigorous framework for generative compressed sensing in Hilbert spaces. We extend the notion of local coherence in an infinite-dimensional setting, to derive optimal, resolution-independent sampling distributions. Thanks to a generalization of the Restricted Isometry Property, we show that stable recovery holds when the number of measurements is proportional to the prior's intrinsic dimension (up to logarithmic factors), independent of the ambient dimension. Finally, numerical experiments on the Darcy flow equation validate our theoretical findings and demonstrate that in severely undersampled regimes, employing lower-resolution generators acts as an implicit regularizer, improving reconstruction stability.
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