# Non-Iterative Recovery from Nonlinear Observations using Generative Models

arXiv: https://arxiv.org/abs/2205.15749 · HTML: https://mlsanity.com/paper/2205.15749/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- Builds on 33 of its 71 references that are in the map.
- First posted on arXiv 31 May 2022.

## Builds on (most cited first, 30 of 33)

- [Deep Learning Face Attributes in the Wild](https://mlsanity.com/paper/1411.7766/) (2014)
- [Improved Training of Wasserstein GANs](https://mlsanity.com/paper/1704.00028/) (2017)
- [Unsupervised Representation Learning with Deep Convolutional Generative Adversarial Networks](https://mlsanity.com/paper/1511.06434/) (2015)
- [Image Super-Resolution via Iterative Refinement](https://mlsanity.com/paper/2104.07636/) (2021)
- [Alias-Free Generative Adversarial Networks](https://mlsanity.com/paper/2106.12423/) (2021)
- [PULSE: Self-Supervised Photo Upsampling via Latent Space Exploration of Generative Models](https://mlsanity.com/paper/2003.03808/) (2020)
- [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)
- [Phase Retrieval via Wirtinger Flow: Theory and Algorithms](https://mlsanity.com/paper/1407.1065/) (2014)
- [Deep Decoder: Concise Image Representations from Untrained Non-convolutional Networks](https://mlsanity.com/paper/1810.03982/) (2018)
- [Stabilizing Training of Generative Adversarial Networks through Regularization](https://mlsanity.com/paper/1705.09367/) (2017)
- [Invertible generative models for inverse problems: mitigating representation error and dataset bias](https://mlsanity.com/paper/1905.11672/) (2019)
- [Phase Retrieval Under a Generative Prior](https://mlsanity.com/paper/1807.04261/) (2018)
- [One Network to Solve Them All --- Solving Linear Inverse Problems using Deep Projection Models](https://mlsanity.com/paper/1703.09912/) (2017)
- [Solving Linear Inverse Problems Using GAN Priors: An Algorithm with Provable Guarantees](https://mlsanity.com/paper/1802.08406/) (2018)
- [Compressed Sensing with Deep Image Prior and Learned Regularization](https://mlsanity.com/paper/1806.06438/) (2018)
- [Deep Compressed Sensing](https://mlsanity.com/paper/1905.06723/) (2019)
- [Instance-Optimal Compressed Sensing via Posterior Sampling](https://mlsanity.com/paper/2106.11438/) (2021)
- [Global Guarantees for Enforcing Deep Generative Priors by Empirical Risk](https://mlsanity.com/paper/1705.07576/) (2017)
- [Modeling Sparse Deviations for Compressed Sensing using Generative Models](https://mlsanity.com/paper/1807.01442/) (2018)
- [Robust Compressed Sensing using Generative Models](https://mlsanity.com/paper/2006.09461/) (2020)
- [An Introductory Guide to Fano's Inequality with Applications in Statistical Estimation](https://mlsanity.com/paper/1901.00555/) (2019)
- [Information-Theoretic Lower Bounds for Compressive Sensing with Generative Models](https://mlsanity.com/paper/1908.10744/) (2019)
- [Sample Complexity Bounds for 1-bit Compressive Sensing and Binary Stable Embeddings with Generative Priors](https://mlsanity.com/paper/2002.01697/) (2020)
- [The Generalized Lasso with Nonlinear Observations and Generative Priors](https://mlsanity.com/paper/2006.12415/) (2020)
- [Provable Compressed Sensing with Generative Priors via Langevin Dynamics](https://mlsanity.com/paper/2102.12643/) (2021)
- [Towards Sample-Optimal Compressive Phase Retrieval with Sparse and Generative Priors](https://mlsanity.com/paper/2106.15358/) (2021)
- [Generative Principal Component Analysis](https://mlsanity.com/paper/2203.09693/) (2022)
- [Robust One-Bit Recovery via ReLU Generative Networks: Near-Optimal Statistical Rate and Global Landscape Analysis](https://mlsanity.com/paper/1908.05368/) (2019)
- [Robust 1-bit Compressive Sensing with Partial Gaussian Circulant Matrices and Generative Priors](https://mlsanity.com/paper/2108.03570/) (2021)

## Built on by (most cited first, 9 of 9)

- [Theoretical Perspectives on Deep Learning Methods in Inverse Problems](https://mlsanity.com/paper/2206.14373/) (2022)
- [Quantized Compressed Sensing with Score-based Generative Models](https://mlsanity.com/paper/2211.13006/) (2022)
- [CS4ML: A general framework for active learning with arbitrary data based on Christoffel functions](https://mlsanity.com/paper/2306.00945/) (2023)
- [A coherence parameter characterizing generative compressed sensing with Fourier measurements](https://mlsanity.com/paper/2207.09340/) (2022)
- [Misspecified Phase Retrieval with Generative Priors](https://mlsanity.com/paper/2210.05571/) (2022)
- [A Unified Framework for Uniform Signal Recovery in Nonlinear Generative Compressed Sensing](https://mlsanity.com/paper/2310.03758/) (2023)
- [QCM-SGM+: Improved Quantized Compressed Sensing With Score-Based Generative Models](https://mlsanity.com/paper/2302.00919/) (2023)
- [Solving Quadratic Systems with Full-Rank Matrices Using Sparse or Generative Priors](https://mlsanity.com/paper/2309.09032/) (2023)
- [Learning Single Index Models with Diffusion Priors](https://mlsanity.com/paper/2505.21135/) (2025)

## Abstract (arXiv)

In this paper, we aim to estimate the direction of an underlying signal from its nonlinear observations following the semi-parametric single index model (SIM). Unlike conventional compressed sensing where the signal is assumed to be sparse, we assume that the signal lies in the range of an L-Lipschitz continuous generative model with bounded k-dimensional inputs. This is mainly motivated by the tremendous success of deep generative models in various real applications. Our reconstruction method is non-iterative (though approximating the projection step may use an iterative procedure) and highly efficient, and it is shown to attain the near-optimal statistical rate of order √(k log L)/m, where m is the number of measurements. We consider two specific instances of the SIM, namely noisy 1-bit and cubic measurement models, and perform experiments on image datasets to demonstrate the efficacy of our method. In particular, for the noisy 1-bit measurement model, we show that our non-iterative method significantly outperforms a state-of-the-art iterative method in terms of both accuracy and efficiency.