# Signal Recovery with Non-Expansive Generative Network Priors

arXiv: https://arxiv.org/abs/2204.13599 · HTML: https://mlsanity.com/paper/2204.13599/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 25 of its 42 references that are in the map.
- First posted on arXiv 24 Apr 2022.

## Builds on (most cited first, 25 of 25)

- [Analyzing and Improving the Image Quality of StyleGAN](https://mlsanity.com/paper/1912.04958/) (2019)
- [Alias-Free Generative Adversarial Networks](https://mlsanity.com/paper/2106.12423/) (2021)
- [Image2StyleGAN: How to Embed Images Into the StyleGAN Latent Space?](https://mlsanity.com/paper/1904.03189/) (2019)
- [PULSE: Self-Supervised Photo Upsampling via Latent Space Exploration of Generative Models](https://mlsanity.com/paper/2003.03808/) (2020)
- [Generative Visual Manipulation on the Natural Image Manifold](https://mlsanity.com/paper/1609.03552/) (2016)
- [Compressed Sensing using Generative Models](https://mlsanity.com/paper/1703.03208/) (2017)
- [Deep Learning Techniques for Inverse Problems in Imaging](https://mlsanity.com/paper/2005.06001/) (2020)
- [Nonconvex Optimization Meets Low-Rank Matrix Factorization: An Overview](https://mlsanity.com/paper/1809.09573/) (2018)
- [No Spurious Local Minima in Nonconvex Low Rank Problems: A Unified Geometric Analysis](https://mlsanity.com/paper/1704.00708/) (2017)
- [Matrix Completion has No Spurious Local Minimum](https://mlsanity.com/paper/1605.07272/) (2016)
- [Deep Decoder: Concise Image Representations from Untrained Non-convolutional Networks](https://mlsanity.com/paper/1810.03982/) (2018)
- [A Geometric Analysis of Phase Retrieval](https://mlsanity.com/paper/1602.06664/) (2016)
- [Phase Retrieval Under a Generative Prior](https://mlsanity.com/paper/1807.04261/) (2018)
- [Complete Dictionary Recovery over the Sphere I: Overview and the Geometric Picture](https://mlsanity.com/paper/1511.03607/) (2015)
- [Stochastic seismic waveform inversion using generative adversarial networks as a geological prior](https://mlsanity.com/paper/1806.03720/) (2018)
- [Global Guarantees for Enforcing Deep Generative Priors by Empirical Risk](https://mlsanity.com/paper/1705.07576/) (2017)
- [Inverting Deep Generative models, One layer at a time](https://mlsanity.com/paper/1906.07437/) (2019)
- [A Provably Convergent Scheme for Compressive Sensing under Random Generative Priors](https://mlsanity.com/paper/1812.04176/) (2018)
- [Global Guarantees for Blind Demodulation with Generative Priors](https://mlsanity.com/paper/1905.12576/) (2019)
- [Constant-Expansion Suffices for Compressed Sensing with Generative Priors](https://mlsanity.com/paper/2006.04237/) (2020)
- [When and How Can Deep Generative Models be Inverted?](https://mlsanity.com/paper/2006.15555/) (2020)
- [Nonsmooth Analysis and Optimization](https://mlsanity.com/paper/1708.04180/) (2017)
- [Rate-Optimal Denoising with Deep Neural Networks](https://mlsanity.com/paper/1805.08855/) (2018)
- [Surfing: Iterative optimization over incrementally trained deep networks](https://mlsanity.com/paper/1907.08653/) (2019)
- [Compressive Phase Retrieval: Optimal Sample Complexity with Deep Generative Priors](https://mlsanity.com/paper/2008.10579/) (2020)

## Built on by (most cited first, 1 of 1)

- [Theoretical Perspectives on Deep Learning Methods in Inverse Problems](https://mlsanity.com/paper/2206.14373/) (2022)

## Abstract (arXiv)

We study compressive sensing with a deep generative network prior. Initial theoretical guarantees for efficient recovery from compressed linear measurements have been developed for signals in the range of a ReLU network with Gaussian weights and logarithmic expansivity: that is when each layer is larger than the previous one by a logarithmic factor. It was later shown that constant expansivity is sufficient for recovery. It has remained open whether the expansivity can be relaxed, allowing for networks with contractive layers (as often the case of real generators). In this work we answer this question, proving that a signal in the range of a Gaussian generative network can be recovered from few linear measurements provided that the width of the layers is proportional to the input layer size (up to log factors). This condition allows the generative network to have contractive layers. Our result is based on showing that Gaussian matrices satisfy a matrix concentration inequality which we term Range Restricted Weight Distribution Condition (R2WDC) and that weakens the Weight Distribution Condition (WDC) upon which previous theoretical guarantees were based. The WDC has also been used to analyze other signal recovery problems with generative network priors. By replacing the WDC with the R2WDC, we are able to extend previous results for signal recovery with expansive generative network priors to non-expansive ones. We discuss these extensions for phase retrieval, denoising, and spiked matrix recovery.