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.
By Zhaoming Li, Paul Hand
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.
By Jing Ma, Yujia Wu, Zhaoqiang Liu
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.
By Roman Pavelkin, Luis A. Zavala-Mondragon, Christiaan G. A. Viviers, Fons van der Sommen
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.
By Sean Gunn, Jorio Cocola, Oliver De Candido, Vaggos Chatziafratis, Paul Hand
arXiv:2509. 15026v2 Announce Type: replace-cross Abstract: We seek to recover an unknown signal from nonlinear amplitude-only measurements, a challenging inverse problem.
By Stanislas Ducotterd, Zhiyuan Hu, Michael Unser, Jonathan Dong
arXiv:2610. 25116v2 Announce Type: replace Abstract: We study positive, bias-free, one-hidden-layer ReLU networks under standard Gaussian inputs.
By Zhijie Zheng
arXiv:2604. 20219v2 Announce Type: replace Abstract: Depth is widely viewed as a central contributor to the success of deep neural networks, whereas standard neural network approximation theory typically provides guarantees only for the final output and leaves the role of intermediate layers largely unclear.
By Shijun Zhang, Zuowei Shen, Yuesheng Xu
arXiv:2509.01809v2 Announce Type: replace-cross
Abstract: We consider the problem of support recovery for sparse binary signals from noisy linear measurements. For sparse Gaussian measurement matrice...
By Youssef Chaabouni, David Gamarnik
arXiv:2607. 16676v1 Announce Type: cross Abstract: How deep does a graph neural network need to be on a sparse graph?
By Aseem Raj Baranwal
arXiv:2608. 11479v1 Announce Type: new Abstract: We establish convergence guarantees of gradient descent for general feedforward neural networks of arbitrary width or depth, with no special requirements on the initialization or dataset.
By Siqiao Mu, Diego Klabjan
arXiv:2610.22693v1 Announce Type: cross
Abstract: We study uniform approximation of the H\"older ball $\mathcal H_1^s([0,1]^D)$ by fully connected ReLU networks. Except in the one-dimensional Lipschi...
By Thomas Nagler
arXiv:2610.02253v1 Announce Type: new
Abstract: The universal approximation property of dropout neural networks does not by itself describe the network size required for an accurate random realizatio...
By Jia-He Yao