# Global Guarantees for Enforcing Deep Generative Priors by Empirical Risk

arXiv: https://arxiv.org/abs/1705.07576 · HTML: https://mlsanity.com/paper/1705.07576/

## Key facts

- Cited by 37 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 22 May 2017.

## Built on by (most cited first, 30 of 37)

- [GAN Inversion: A Survey](https://mlsanity.com/paper/2101.05278/) (2021)
- [Robust Compressed Sensing MRI with Deep Generative Priors](https://mlsanity.com/paper/2108.01368/) (2021)
- [Intermediate Layer Optimization for Inverse Problems using Deep Generative Models](https://mlsanity.com/paper/2102.07364/) (2021)
- [Instance-Optimal Compressed Sensing via Posterior Sampling](https://mlsanity.com/paper/2106.11438/) (2021)
- [Scaling and Scalability: Provable Nonconvex Low-Rank Tensor Estimation from Incomplete Measurements](https://mlsanity.com/paper/2104.14526/) (2021)
- [Bringing Your Own View: Graph Contrastive Learning without Prefabricated Data Augmentations](https://mlsanity.com/paper/2201.01702/) (2022)
- [Solving Inverse Problems by Joint Posterior Maximization with Autoencoding Prior](https://mlsanity.com/paper/2103.01648/) (2021)
- [Regularising Inverse Problems with Generative Machine Learning Models](https://mlsanity.com/paper/2107.11191/) (2021)
- [Theoretical Perspectives on Deep Learning Methods in Inverse Problems](https://mlsanity.com/paper/2206.14373/) (2022)
- [Provable Compressed Sensing with Generative Priors via Langevin Dynamics](https://mlsanity.com/paper/2102.12643/) (2021)
- [Convergence and Alignment of Gradient Descent with Random Backpropagation Weights](https://mlsanity.com/paper/2106.06044/) (2021)
- [Score-Guided Intermediate Layer Optimization: Fast Langevin Mixing for Inverse Problems](https://mlsanity.com/paper/2206.09104/) (2022)
- [Analysis of Diffusion Models for Manifold Data](https://mlsanity.com/paper/2502.04339/) (2025)
- [Non-Iterative Recovery from Nonlinear Observations using Generative Models](https://mlsanity.com/paper/2205.15749/) (2022)
- [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)
- [Optimization for Amortized Inverse Problems](https://mlsanity.com/paper/2210.13983/) (2022)
- [Generative adversarial networks with physical sound field priors](https://mlsanity.com/paper/2308.00426/) (2023)
- [Iso-Riemannian Optimization on Learned Data Manifolds](https://mlsanity.com/paper/2510.21033/) (2025)
- [Denoising Generalized Expectation-Consistent Approximation for MR Image Recovery](https://mlsanity.com/paper/2206.05049/) (2022)
- [How many measurements are enough? Bayesian recovery in inverse problems with general distributions](https://mlsanity.com/paper/2505.10630/) (2025)
- [Non-Convex Compressed Sensing with Training Data](https://mlsanity.com/paper/2101.08310/) (2021)
- [Provably Convergent Algorithms for Solving Inverse Problems Using Generative Models](https://mlsanity.com/paper/2105.06371/) (2021)
- [Learning Generative Prior with Latent Space Sparsity Constraints](https://mlsanity.com/paper/2105.11956/) (2021)
- [Signal Recovery with Non-Expansive Generative Network Priors](https://mlsanity.com/paper/2204.13599/) (2022)
- [On the Fine-Grained Hardness of Inverting Generative Models](https://mlsanity.com/paper/2309.05795/) (2023)
- [Riemannian AmbientFlow: Towards Simultaneous Manifold Learning and Generative Modeling from Corrupted Data](https://mlsanity.com/paper/2601.18728/) (2026)
- [Riemannian Archetypal Analysis: Interpretable non-linear data analysis on deformed star distributions](https://mlsanity.com/paper/2605.24113/) (2026)
- [Training Deep Neural Networks via Branch-and-Bound](https://mlsanity.com/paper/2104.01730/) (2021)

## Abstract (arXiv)

We examine the theoretical properties of enforcing priors provided by generative deep neural networks via empirical risk minimization. In particular we consider two models, one in which the task is to invert a generative neural network given access to its last layer and another in which the task is to invert a generative neural network given only compressive linear observations of its last layer. We establish that in both cases, in suitable regimes of network layer sizes and a randomness assumption on the network weights, that the non-convex objective function given by empirical risk minimization does not have any spurious stationary points. That is, we establish that with high probability, at any point away from small neighborhoods around two scalar multiples of the desired solution, there is a descent direction. Hence, there are no local minima, saddle points, or other stationary points outside these neighborhoods. These results constitute the first theoretical guarantees which establish the favorable global geometry of these non-convex optimization problems, and they bridge the gap between the empirical success of enforcing deep generative priors and a rigorous understanding of non-linear inverse problems.