# Limitations of Deep Learning for Inverse Problems on Digital Hardware

arXiv: https://arxiv.org/abs/2202.13490 · HTML: https://mlsanity.com/paper/2202.13490/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- Builds on 32 of its 97 references that are in the map.
- First posted on arXiv 28 Feb 2022.

## Builds on (most cited first, 30 of 32)

- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [Towards Deep Learning Models Resistant to Adversarial Attacks](https://mlsanity.com/paper/1706.06083/) (2017)
- [Intriguing properties of neural networks](https://mlsanity.com/paper/1312.6199/) (2013)
- [Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification](https://mlsanity.com/paper/1502.01852/) (2015)
- [DeepFool: a simple and accurate method to fool deep neural networks](https://mlsanity.com/paper/1511.04599/) (2015)
- [Adversarial Examples Are Not Bugs, They Are Features](https://mlsanity.com/paper/1905.02175/) (2019)
- [Robustness May Be at Odds with Accuracy](https://mlsanity.com/paper/1805.12152/) (2018)
- [Distillation as a Defense to Adversarial Perturbations against Deep Neural Networks](https://mlsanity.com/paper/1511.04508/) (2015)
- [Threat of Adversarial Attacks on Deep Learning in Computer Vision: A Survey](https://mlsanity.com/paper/1801.00553/) (2018)
- [Learning to See in the Dark](https://mlsanity.com/paper/1805.01934/) (2018)
- [Deep Convolutional Neural Network for Inverse Problems in Imaging](https://mlsanity.com/paper/1611.03679/) (2016)
- [Algorithm Unrolling: Interpretable, Efficient Deep Learning for Signal and Image Processing](https://mlsanity.com/paper/1912.10557/) (2019)
- [Learning a Variational Network for Reconstruction of Accelerated MRI Data](https://mlsanity.com/paper/1704.00447/) (2017)
- [Deep Learning Techniques for Inverse Problems in Imaging](https://mlsanity.com/paper/2005.06001/) (2020)
- [Audio Adversarial Examples: Targeted Attacks on Speech-to-Text](https://mlsanity.com/paper/1801.01944/) (2018)
- [A Deep Cascade of Convolutional Neural Networks for Dynamic MR Image Reconstruction](https://mlsanity.com/paper/1704.02422/) (2017)
- [Image reconstruction by domain transform manifold learning](https://mlsanity.com/paper/1704.08841/) (2017)
- [On instabilities of deep learning in image reconstruction - Does AI come at a cost?](https://mlsanity.com/paper/1902.05300/) (2019)
- [Solving ill-posed inverse problems using iterative deep neural networks](https://mlsanity.com/paper/1704.04058/) (2017)
- [Explainable Deep Learning: A Field Guide for the Uninitiated](https://mlsanity.com/paper/2004.14545/) (2020)
- [A Survey of Neuromorphic Computing and Neural Networks in Hardware](https://mlsanity.com/paper/1705.06963/) (2017)
- [The Modern Mathematics of Deep Learning](https://mlsanity.com/paper/2105.04026/) (2021)
- [The gap between theory and practice in function approximation with deep neural networks](https://mlsanity.com/paper/2001.07523/) (2020)
- [Can stable and accurate neural networks be computed? -- On the barriers of deep learning and Smale's 18th problem](https://mlsanity.com/paper/2101.08286/) (2021)
- [A Deep Learning Approach to Structured Signal Recovery](https://mlsanity.com/paper/1508.04065/) (2015)
- [The troublesome kernel -- On hallucinations, no free lunches and the accuracy-stability trade-off in inverse problems](https://mlsanity.com/paper/2001.01258/) (2020)
- [Learning The Invisible: A Hybrid Deep Learning-Shearlet Framework for Limited Angle Computed Tomography](https://mlsanity.com/paper/1811.04602/) (2018)
- [The mathematics of adversarial attacks in AI -- Why deep learning is unstable despite the existence of stable neural networks](https://mlsanity.com/paper/2109.06098/) (2021)
- [The extended Smale's 9th problem -- On computational barriers and paradoxes in estimation, regularisation, computer-assisted proofs and learning](https://mlsanity.com/paper/2110.15734/) (2021)
- [A Rate-Distortion Framework for Explaining Black-box Model Decisions](https://mlsanity.com/paper/2110.08252/) (2021)

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

- [NESTANets: Stable, accurate and efficient neural networks for analysis-sparse inverse problems](https://mlsanity.com/paper/2203.00804/) (2022)
- [Computability of Optimizers](https://mlsanity.com/paper/2301.06148/) (2023)
- [Deep neural networks can stably solve high-dimensional, noisy, non-linear inverse problems](https://mlsanity.com/paper/2206.00934/) (2022)
- [Limitations of neural network training due to numerical instability of backpropagation](https://mlsanity.com/paper/2210.00805/) (2022)
- [Mathematical Algorithm Design for Deep Learning under Societal and Judicial Constraints: The Algorithmic Transparency Requirement](https://mlsanity.com/paper/2401.10310/) (2024)
- [Deep-Unrolling Multidimensional Harmonic Retrieval Algorithms on Neuromorphic Hardware](https://mlsanity.com/paper/2412.04008/) (2024)
- [Reliable AI: Does the Next Generation Require Quantum Computing?](https://mlsanity.com/paper/2307.01301/) (2023)
- [Tighter Learning Guarantees on Digital Computers via Concentration of Measure on Finite Spaces](https://mlsanity.com/paper/2402.05576/) (2024)
- [Computability of Classification and Deep Learning: From Theoretical Limits to Practical Feasibility through Quantization](https://mlsanity.com/paper/2408.06212/) (2024)

## Abstract (arXiv)

Deep neural networks have seen tremendous success over the last years. Since the training is performed on digital hardware, in this paper, we analyze what actually can be computed on current hardware platforms modeled as Turing machines, which would lead to inherent restrictions of deep learning. For this, we focus on the class of inverse problems, which, in particular, encompasses any task to reconstruct data from measurements. We prove that finite-dimensional inverse problems are not Banach-Mazur computable for small relaxation parameters. Even more, our results introduce a lower bound on the accuracy that can be obtained algorithmically.