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Learning with known operators reduces maximum error bounds


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 show experimentally that known operators reduce the number of free parameters. We apply this approach to various tasks ranging from computed tomography 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.

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Fig. 1: Schematic of the idea of known operator learning.
Fig. 2: Deep learning computed tomography.
Fig. 3: Improved interpretability in deep networks.
Fig. 4: Architecture of Frangi-Net over eight scales σ.
Fig. 5: Classical analytical rebinning versus derived neural networks.
Fig. 6: Towards operator discovery and sequence analysis.

Data availability

All data in this publication are publicly available. The experiments for deep learning computed tomography use data from the low-dose CT challenge39. The section on learning from heuristic algorithms uses the DRIVE database40. The data for the section on deriving networks is available in a Code Ocean Capsule available at

Code availability

The code and data for this Article, along with an accompanying computational environment, are available and executable online as a Code Ocean Capsule. Experiments in the section on deep learning computed tomography can be found at The code on learning vesselness in the section on learning from heuristic algorithms is published at Code for the section on deriving networks is available at Code capsules for experiments in the sections on deep learning computed tomography and deriving networks were implemented using the open source framework PYRO-NN44.


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The research leading to these results has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (ERC grant no. 810316).

Author information




A.K.M. is the main author of the paper and is responsible for writing the manuscript, theoretical analysis and experimental design. C.S. and B.S. contributed to writing the section about deriving networks as well as its supporting experiments. T.W. and M.H. helped with writing the section on deep learning computed tomography and performed the experiments reported in that section. F.S. contributed to the mathematical analysis and the writing thereof. W.F. conducted the experiments supporting the section on learning from heuristic algorithms and contributed to their description. L.M., L.K. and S.C. contributed to the experimental design and the writing of the manuscript.

Corresponding author

Correspondence to Andreas K. Maier.

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Maier, A.K., Syben, C., Stimpel, B. et al. Learning with known operators reduces maximum error bounds. Nat Mach Intell 1, 373–380 (2019).

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