# Learning with Known Operators reduces Maximum Training Error Bounds

arXiv: https://arxiv.org/abs/1907.01992 · HTML: https://mlsanity.com/paper/1907.01992/

## Key facts

- Cited by 25 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 3 Jul 2019.

## Built on by (most cited first, 25 of 25)

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- [Known Operator Learning and Hybrid Machine Learning in Medical Imaging -- A Review of the Past, the Present, and the Future](https://mlsanity.com/paper/2108.04543/) (2021)
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- [A Scoping Review of Machine Learning Applications in Power System Protection and Disturbance Management](https://mlsanity.com/paper/2509.09053/) (2025)
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- [An End-To-End-Trainable Iterative Network Architecture for Accelerated Radial Multi-Coil 2D Cine MR Image Reconstruction](https://mlsanity.com/paper/2102.00783/) (2021)
- [Trainable Joint Bilateral Filters for Enhanced Prediction Stability in Low-dose CT](https://mlsanity.com/paper/2207.07368/) (2022)
- [Data Models for Dataset Drift Controls in Machine Learning With Optical Images](https://mlsanity.com/paper/2211.02578/) (2022)
- [Artifact Reduction in 3D and 4D Cone-beam Computed Tomography Images with Deep Learning -- A Review](https://mlsanity.com/paper/2403.18565/) (2024)
- [Machine learning in business process management: A systematic literature review](https://mlsanity.com/paper/2405.16396/) (2024)
- [Noise2Contrast: Multi-Contrast Fusion Enables Self-Supervised Tomographic Image Denoising](https://mlsanity.com/paper/2212.04832/) (2022)
- [Learning the Update Operator for 2D/3D Image Registration](https://mlsanity.com/paper/2102.02861/) (2021)
- [Robust partial Fourier reconstruction for diffusion-weighted imaging using a recurrent convolutional neural network](https://mlsanity.com/paper/2105.09378/) (2021)
- [Physically Inspired Constraint for Unsupervised Regularized Ultrasound Elastography](https://mlsanity.com/paper/2206.02225/) (2022)
- [Lateral Strain Imaging using Self-supervised and Physically Inspired Constraints in Unsupervised Regularized Elastography](https://mlsanity.com/paper/2212.08740/) (2022)
- [Convolutional Dictionary Learning by End-To-End Training of Iterative Neural Networks](https://mlsanity.com/paper/2206.04447/) (2022)
- [A Deep Risk Estimator for Known Operator Learning](https://mlsanity.com/paper/2605.08517/) (2026)
- [SEMPAI: a Self-Enhancing Multi-Photon Artificial Intelligence for prior-informed assessment of muscle function and pathology](https://mlsanity.com/paper/2210.16273/) (2022)
- [Active Inference-Based Optimization of Discriminative Neural Network Classifiers](https://mlsanity.com/paper/2306.02447/) (2023)
- [Quantitative Analysis of Molecular Transport in the Extracellular Space Using Physics-Informed Neural Network](https://mlsanity.com/paper/2401.12435/) (2024)
- [DiffRenderGAN: Addressing Training Data Scarcity in Deep Segmentation Networks for Quantitative Nanomaterial Analysis through Differentiable Rendering and Generative Modelling](https://mlsanity.com/paper/2502.09477/) (2025)

## Abstract (arXiv)

We describe an approach for incorporating prior knowledge into machine learning algorithms. We aim at applications in physics and signal processing in which we know that certain operations must be embedded into the algorithm. Any operation that allows computation of a gradient or sub-gradient towards its inputs is suited for our framework. We derive a maximal error bound for deep nets that demonstrates that inclusion of prior knowledge results in its reduction. Furthermore, we also show experimentally that known operators reduce the number of free parameters. We apply this approach to various tasks ranging from CT image reconstruction over vessel segmentation to the derivation of previously unknown imaging algorithms. As such the concept is widely applicable for many researchers in physics, imaging, and signal processing. We assume that our analysis will support further investigation of known operators in other fields of physics, imaging, and signal processing.