Full-Model Optimality for Tunable Linear Generative Priors in Compressed Sensing
Abstract
Generative models have been studied experimentally and theoretically as priors for inverse problems such as compressed sensing. Recent work by Gunn et al. studied the use of generative priors with tunable complexity, where a family of generative priors with varying complexity is maintained and a specific complexity can be selected at inversion time. They demonstrated that lower reconstruction errors can be experimentally attained for a variety of inverse problems by appropriately tuning the complexity of the generative prior. In the present paper, we establish theory for compressed sensing in the setting of a tunable family of linear generative priors naturally related through their singular value decompositions. We prove that in noiseless Gaussian compressed sensing, the full-dimensional linear prior attains the minimum expected reconstruction error over the entire family of linear priors. Thus, in this idealized linear noiseless setting, tuning to a lower-complexity prior does not improve the expected reconstruction error. This result is in contract to the behavior of denoising, where lower complexity priors attain lower reconstruction errors due to a standard bias-variance tradeoff. This result indicates that the experimental benefits of tunability in compressed sensing with neural network priors arises due to nonlinearities in the generative models.
Related
- Superfast Spectral Compressed Sensing with Quadratic Convergence
- Recovery Guarantees for Posterior Sampling of One-Bit Compressed Sensing
- Phase Transition in Binary Compressed Sensing via Annealing with Adaptive Regularization
- Compressed Sensing with Quantized Tensor Trains (QTTs)
- Lifted Restricted Isometry Property and Fast Algorithm for Spectral Compressed Sensing With Near-Optimal Sample Complexity
BibTeX
@misc{li2026fullmodel,
title = {{Full-Model Optimality for Tunable Linear Generative Priors in Compressed Sensing}},
author = {Zhaoming Li and Paul Hand},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.02790},
eprint = {2609.02790},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.02790},
}
RIS
TY - UNPB
TI - Full-Model Optimality for Tunable Linear Generative Priors in Compressed Sensing
AU - Zhaoming Li
AU - Paul Hand
PY - 2026
DA - 2026/09/02
PB - arXiv
DO - 10.48550/arxiv.2609.02790
UR - https://arxiv.org/abs/2609.02790
AB - Generative models have been studied experimentally and theoretically as priors for inverse problems such as compressed sensing. Recent work by Gunn et al. studied the use of generative priors with tunable complexity, where a family of generative priors with varying complexity is maintained and a specific complexity can be selected at inversion time. They demonstrated that lower reconstruction errors can be experimentally attained for a variety of inverse problems by appropriately tuning the complexity of the generative prior. In the present paper, we establish theory for compressed sensing in the setting of a tunable family of linear generative priors naturally related through their singular value decompositions. We prove that in noiseless Gaussian compressed sensing, the full-dimensional linear prior attains the minimum expected reconstruction error over the entire family of linear priors. Thus, in this idealized linear noiseless setting, tuning to a lower-complexity prior does not improve the expected reconstruction error. This result is in contract to the behavior of denoising, where lower complexity priors attain lower reconstruction errors due to a standard bias-variance tradeoff. This result indicates that the experimental benefits of tunability in compressed sensing with neural network priors arises due to nonlinearities in the generative models.
ER -