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.
Figures & tables
Figure 1. Comparison between the original maximum-magnitude based coherence estimator and the self-difference based estimator. Although the latter is theoretically better aligned with the definition of local coherence, the former yields superior empirical performance (model trained and with output in 64 x 64 resolution).
Figure 2. Reconstruction error versus sampling rate for uniform and adaptive Fourier sampling.
Figure 3. Reconstruction examples comparing uniform and adaptive sampling at different rates (model trained on 64 x 64 resolution).
Figure 4. Recovery error across different training and output discretizations.
Figure 5. Example reconstructions for varying output resolutions (model trained on 64 x 64 resolution)
Figure 6. Comparison between our proposed method and classical compressed sensing (CS).