# The inverse problem for neural networks

arXiv: https://arxiv.org/abs/2308.14093 · HTML: https://mlsanity.com/paper/2308.14093/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 17 of its 31 references that are in the map.
- First posted on arXiv 27 Aug 2023.

## Builds on (most cited first, 17 of 17)

- [Explaining and Harnessing Adversarial Examples](https://mlsanity.com/paper/1412.6572/) (2014)
- [Deep Inside Convolutional Networks: Visualising Image Classification Models and Saliency Maps](https://mlsanity.com/paper/1312.6034/) (2013)
- [Visualizing and Understanding Convolutional Networks](https://mlsanity.com/paper/1311.2901/) (2013)
- [Understanding Deep Image Representations by Inverting Them](https://mlsanity.com/paper/1412.0035/) (2014)
- [On the Number of Linear Regions of Deep Neural Networks](https://mlsanity.com/paper/1402.1869/) (2014)
- [Formal Security Analysis of Neural Networks using Symbolic Intervals](https://mlsanity.com/paper/1804.10829/) (2018)
- [When Explanations Lie: Why Many Modified BP Attributions Fail](https://mlsanity.com/paper/1912.09818/) (2019)
- [Provably Bounding Neural Network Preimages](https://mlsanity.com/paper/2302.01404/) (2023)
- [JuliaReach: a Toolbox for Set-Based Reachability](https://mlsanity.com/paper/1901.10736/) (2019)
- [Verification of Neural-Network Control Systems by Integrating Taylor Models and Zonotopes](https://mlsanity.com/paper/2112.09197/) (2021)
- [Open- and Closed-Loop Neural Network Verification using Polynomial Zonotopes](https://mlsanity.com/paper/2207.02715/) (2022)
- [SyReNN: A Tool for Analyzing Deep Neural Networks](https://mlsanity.com/paper/2101.03263/) (2021)
- [Backward Reachability Analysis for Neural Feedback Loops](https://mlsanity.com/paper/2204.08319/) (2022)
- [Neural Network Compression of ACAS Xu Early Prototype is Unsafe: Closed-Loop Verification through Quantized State Backreachability](https://mlsanity.com/paper/2201.06626/) (2022)
- [A Hybrid Partitioning Strategy for Backward Reachability of Neural Feedback Loops](https://mlsanity.com/paper/2210.07918/) (2022)
- [DRIP: Domain Refinement Iteration with Polytopes for Backward Reachability Analysis of Neural Feedback Loops](https://mlsanity.com/paper/2212.04646/) (2022)
- [LazySets.jl: Scalable Symbolic-Numeric Set Computations](https://mlsanity.com/paper/2110.01711/) (2021)

## Abstract (arXiv)

We study the problem of computing the preimage of a set under a neural network with piecewise-affine activation functions. We recall an old result that the preimage of a polyhedral set is again a union of polyhedral sets and can be effectively computed. We show several applications of computing the preimage for analysis and interpretability of neural networks.