Exact signal recovery from generative priors with sigmoid output activation
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The paper investigates compressed sensing using a family of tunable linear generative priors linked through their singular value decompositions. It proves that in noiseless Gaussian compressed sensing, the full-dimensional linear prior yields the lowest expected reconstruction error across the entire family, meaning lower-complexity priors do not improve performance in this idealized setting. This contrasts with denoising, where lower complexity priors can reduce error due to bias‑variance tradeoffs, suggesting that the experimental gains seen with neural network priors stem from their nonlinearities.
The paper investigates the sample complexity of noisy one‑bit compressed sensing for signals drawn from a prior distribution. By linking the prior’s effective distributional complexity to its approximate covering number, the authors prove that posterior sampling can recover signals accurately when the number of measurements scales with the logarithm of this covering number, modulo a one‑bit separation gap. They also show that the method remains reliable under learned prior mismatch measured in Wasserstein distance, provide a matching lower bound, and demonstrate practical performance using a diffusion‑based plug‑and‑play algorithm on FFHQ and ImageNet.
arXiv:2606. 00078v1 Announce Type: cross Abstract: Numerous modern applications in signal processing and medical imaging necessitate acquiring high-dimensional signals under tight resource constraints.
The paper introduces tunable latent priors for diffusion models, normalizing flows, and variational autoencoders using nested dropout. These priors allow the latent dimensionality to adapt to each inverse problem, reducing reconstruction errors compared to fixed-complexity models across tasks such as compressed sensing, inpainting, denoising, and phase retrieval. In linear denoising, the authors derive the optimal latent complexity in closed form, linking it to noise level and signal spectrum.
arXiv:2509. 15026v2 Announce Type: replace-cross Abstract: We seek to recover an unknown signal from nonlinear amplitude-only measurements, a challenging inverse problem.
arXiv:2610. 25116v2 Announce Type: replace Abstract: We study positive, bias-free, one-hidden-layer ReLU networks under standard Gaussian inputs.