  Following the great success of Machine Learning (ML), especially Deep Neural
Networks (DNNs), in many research domains in 2010s, several learning-based
approaches were proposed for detection in large inverse linear problems, e.g.,
massive MIMO systems. The main motivation behind is that the complexity of
Maximum A-Posteriori (MAP) detection grows exponentially with system
dimensions. Instead of using DNNs, essentially being a black-box in its most
basic form, we take a slightly different approach and introduce a probabilistic
Continuous relaxation of disCrete variables to MAP detection. Enabling close
approximation and continuous optimization, we derive an iterative detection
algorithm: ConCrete MAP Detection (CMD). Furthermore, by extending CMD to the
idea of deep unfolding, we allow for (online) optimization of a small number of
parameters to different working points while limiting complexity. In contrast
to recent DNN-based approaches, we select the optimization criterion and output
of CMD based on information theory and are thus able to learn approximate
probabilities of the individual optimal detector. This is crucial for soft
decoding in today's communication systems. Numerical simulation results in MIMO
systems reveal CMD to feature a promising performance complexity trade-off
compared to SotA. Notably, we demonstrate CMD's soft outputs to be reliable for
decoders.
