# The Generalized Lasso with Nonlinear Observations and Generative Priors

arXiv: https://arxiv.org/abs/2006.12415 · HTML: https://mlsanity.com/paper/2006.12415/

## Key facts

- Cited by 15 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 22 Jun 2020.

## Built on by (most cited first, 15 of 15)

- [DiffFit: Unlocking Transferability of Large Diffusion Models via Simple Parameter-Efficient Fine-Tuning](https://mlsanity.com/paper/2304.06648/) (2023)
- [Theoretical Perspectives on Deep Learning Methods in Inverse Problems](https://mlsanity.com/paper/2206.14373/) (2022)
- [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)
- [Non-Iterative Recovery from Nonlinear Observations using Generative Models](https://mlsanity.com/paper/2205.15749/) (2022)
- [Robust 1-bit Compressive Sensing with Partial Gaussian Circulant Matrices and Generative Priors](https://mlsanity.com/paper/2108.03570/) (2021)
- [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)
- [Integrating Intermediate Layer Optimization and Projected Gradient Descent for Solving Inverse Problems with Diffusion Models](https://mlsanity.com/paper/2505.20789/) (2025)
- [Projected Gradient Descent Algorithms for Solving Nonlinear Inverse Problems with Generative Priors](https://mlsanity.com/paper/2209.10093/) (2022)
- [Equivariant Priors for Compressed Sensing with Unknown Orientation](https://mlsanity.com/paper/2206.14069/) (2022)
- [Generalized Eigenvalue Problems with Generative Priors](https://mlsanity.com/paper/2411.01326/) (2024)
- [Just Least Squares: Binary Compressive Sampling with Low Generative Intrinsic Dimension](https://mlsanity.com/paper/2111.14486/) (2021)
- [Learning Single Index Models with Diffusion Priors](https://mlsanity.com/paper/2505.21135/) (2025)
- [Diffusion Model Based Signal Recovery Under 1-Bit Quantization](https://mlsanity.com/paper/2511.12471/) (2025)

## Abstract (arXiv)

In this paper, we study the problem of signal estimation from noisy non-linear measurements when the unknown n-dimensional signal is in the range of an L-Lipschitz continuous generative model with bounded k-dimensional inputs. We make the assumption of sub-Gaussian measurements, which is satisfied by a wide range of measurement models, such as linear, logistic, 1-bit, and other quantized models. In addition, we consider the impact of adversarial corruptions on these measurements. Our analysis is based on a generalized Lasso approach (Plan and Vershynin, 2016). We first provide a non-uniform recovery guarantee, which states that under i.i.d.~Gaussian measurements, roughly Oleft(frackε2log Lright) samples suffice for recovery with an ℓ2-error of ε, and that this scheme is robust to adversarial noise. Then, we apply this result to neural network generative models, and discuss various extensions to other models and non-i.i.d.~measurements. Moreover, we show that our result can be extended to the uniform recovery guarantee under the assumption of a so-called local embedding property, which is satisfied by the 1-bit and censored Tobit models.