Learning with Known Operators reduces Maximum Training 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. Deep neural networks can contain arbitrary mathematical operators, as long as they are derivable. The authors investigate how knowledge about a problem can be incorporated into machine learning through the use of operators that are related to the problem.