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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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 https://doi.org/10.24433/CO.8086142.v241.
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 https://doi.org/10.24433/CO.2164960.v142. The code on learning vesselness in the section on learning from heuristic algorithms is published at https://doi.org/10.24433/CO.5016803.v241. Code for the section on deriving networks is available at https://doi.org/10.24433/CO.8086142.v243. 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).
The authors declare no competing interests.
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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). https://doi.org/10.1038/s42256-019-0077-5
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