## 3 Parameter Learning of Trainable Functions to Maximize the Relevant Information
### Computational Domain Technique
It has to be appreciated that the proposed method from [24; 25] offers excellent bit error rate performance for LDPC decoding. To illustrate this fact, Figure 14 compares the bit error rate performances of the computational domain approach and two lookup table-based information bottleneck decoders for the \((d_{\mathrm{v}},d_{\mathrm{c}})=(3,6)\) regular LDPC code that was already used in Section 2.3.4. For these simulations, all quantized decoders exchanged \(q=4\)-bit messages in the iterative message passing process for their respective decoding. The bit width used internally to represent \(a\in\mathcal{A}\) in the computational domain decoders was 10 bits for the check nodes and variable nodes. The shown results for the computational domain decoder were adopted from [24]. The bit error rate curves refer to data transmission with BPSK modulation over an additive white Gaussian noise channel. The number of decoder iterations was \(i_{\mathrm{max}}=50\) for all investigated decoders. The relatively small node degrees of the applied code allow constructing lookup table-based information bottleneck decoders with and without splitting the node operation into two-input operations. We want to mention that the decoder without splitting has an impractically large memory demand but shall be used as a benchmark here. For the relatively small node degrees of the \((3,6)\) regular LDPC code used here, the lookup tables in the decoder without splitting already had more than 300 million entries [28]. As can be seen, the computational domain decoder performed just as well as a lookup table-based decoder without internal splitting of the node operations. Both quantized decoders effectively reached the performance of the non-quantized belief propagation decoder with double-precision up to a negligible gap over \(E_{b}/N_{0}\). As can also be seen, splitting the node operation of the information bottleneck decoder into a series of two-input operations offered slightly worse performance but yielded much smaller lookup tables in return. The number of lookup table entries used for this decoder was just 384,000 and therefore drastically reduced to a tractable number in comparison with the lookup table-based decoder without two-input splitting of the node operations. Again, we want to stress here that the information bottleneck decoders and the computational domain decoder [24] were entirely constructed offline and used mismatched to the \(E_{b}/N_{0}\) on the channel. For reference, the bit error rate of the well-known min-sum decoder with double-precision is also shown in Figure 14. This decoder was outperformed by all decoders designed with the information bottleneck principle of maximizing the preserved relevant information. Finally, we note that the computational domain approach is well suited for LDPC decoding. Thus far, to the best of our knowledge, studies of computational domain approaches that maximize the relevant information for other applications of information bottleneck signal processing such as the channel estimation studied in Section 2.3.1, cannot be found in the literature.