# Inverting Deep Generative models, One layer at a time

arXiv: https://arxiv.org/abs/1906.07437 · HTML: https://mlsanity.com/paper/1906.07437/

## Key facts

- Cited by 33 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 18 Jun 2019.

## Built on by (most cited first, 30 of 33)

- [GAN Inversion: A Survey](https://mlsanity.com/paper/2101.05278/) (2021)
- [See through Gradients: Image Batch Recovery via GradInversion](https://mlsanity.com/paper/2104.07586/) (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)
- [Unsupervised 3D Shape Completion through GAN Inversion](https://mlsanity.com/paper/2104.13366/) (2021)
- [Stochastic Image Denoising by Sampling from the Posterior Distribution](https://mlsanity.com/paper/2101.09552/) (2021)
- [Instance-Optimal Compressed Sensing via Posterior Sampling](https://mlsanity.com/paper/2106.11438/) (2021)
- [Reconstructing Training Data from Model Gradient, Provably](https://mlsanity.com/paper/2212.03714/) (2022)
- [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)
- [Universal Joint Approximation of Manifolds and Densities by Simple Injective Flows](https://mlsanity.com/paper/2110.04227/) (2021)
- [Single-Model Attribution of Generative Models Through Final-Layer Inversion](https://mlsanity.com/paper/2306.06210/) (2023)
- [On Linear Interpolation in the Latent Space of Deep Generative Models](https://mlsanity.com/paper/2105.03663/) (2021)
- [Privacy Vulnerability of Split Computing to Data-Free Model Inversion Attacks](https://mlsanity.com/paper/2107.06304/) (2021)
- [Continuous Generative Neural Networks: A Wavelet-Based Architecture in Function Spaces](https://mlsanity.com/paper/2205.14627/) (2022)
- [Optimization for Amortized Inverse Problems](https://mlsanity.com/paper/2210.13983/) (2022)
- [Injectivity capacity of ReLU gates](https://mlsanity.com/paper/2410.20646/) (2024)
- [Iso-Riemannian Optimization on Learned Data Manifolds](https://mlsanity.com/paper/2510.21033/) (2025)
- [Equivariant Priors for Compressed Sensing with Unknown Orientation](https://mlsanity.com/paper/2206.14069/) (2022)
- [Deep Joint Source-Channel Coding with Iterative Source Error Correction](https://mlsanity.com/paper/2302.09174/) (2023)
- [Roulette: A Semantic Privacy-Preserving Device-Edge Collaborative Inference Framework for Deep Learning Classification Tasks](https://mlsanity.com/paper/2309.02820/) (2023)
- [Provably Convergent Algorithms for Solving Inverse Problems Using Generative Models](https://mlsanity.com/paper/2105.06371/) (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)
- [ReQuestNet: A Foundational Learning model for Channel Estimation](https://mlsanity.com/paper/2508.08790/) (2025)
- [Riemannian Archetypal Analysis: Interpretable non-linear data analysis on deformed star distributions](https://mlsanity.com/paper/2605.24113/) (2026)
- [The Effects of Invertibility on the Representational Complexity of Encoders in Variational Autoencoders](https://mlsanity.com/paper/2107.04652/) (2021)
- [A Linearly Convergent GAN Inversion-based Algorithm for Reverse Engineering of Deceptions](https://mlsanity.com/paper/2306.04756/) (2023)
- [Beyond Invisibility: Learning Robust Visible Watermarks for Stronger Copyright Protection](https://mlsanity.com/paper/2506.02665/) (2025)

## Abstract (arXiv)

We study the problem of inverting a deep generative model with ReLU activations. Inversion corresponds to finding a latent code vector that explains observed measurements as much as possible. In most prior works this is performed by attempting to solve a non-convex optimization problem involving the generator. In this paper we obtain several novel theoretical results for the inversion problem. We show that for the realizable case, single layer inversion can be performed exactly in polynomial time, by solving a linear program. Further, we show that for multiple layers, inversion is NP-hard and the pre-image set can be non-convex. For generative models of arbitrary depth, we show that exact recovery is possible in polynomial time with high probability, if the layers are expanding and the weights are randomly selected. Very recent work analyzed the same problem for gradient descent inversion. Their analysis requires significantly higher expansion (logarithmic in the latent dimension) while our proposed algorithm can provably reconstruct even with constant factor expansion. We also provide provable error bounds for different norms for reconstructing noisy observations. Our empirical validation demonstrates that we obtain better reconstructions when the latent dimension is large.