# Information Bottleneck Signal Processing and Learning to Maximize Relevant Information for Communication Receivers

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## Abstract

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## 1. Introduction

## 2. The Information Bottleneck Method and Coarsely Quantized Information Bottleneck Signal Processing

#### 2.1. The Information Bottleneck Method

- Conduct a lossy compression of the realizations $y\in \mathcal{Y}$ to a compressed realization $t\in \mathcal{T}$ to yield a compact compressed representation $\mathsf{T}$ of the observation $\mathsf{Y}$. The information theoretical notion of such a compression is the minimization of the compression information I$(\mathsf{Y};\mathsf{T})$ (i.e., the transmission rate, relating to rate–distortion theory).
- While conducting the compression mentioned above, preserve the relevant information $\mathrm{I}(\mathsf{X};\mathsf{T})\le \mathrm{I}(\mathsf{X};\mathsf{Y})$.

#### 2.2. General View on Information Bottleneck Signal Processing for Receiver Design

#### 2.3. An Example of Information Bottleneck Receiver Design with Iterative Detection and Decoding

#### 2.3.1. Information Bottleneck Channel Estimation

#### 2.3.2. Information Bottleneck Detection

#### 2.3.3. Information Bottleneck LDPC Decoder Design

#### 2.3.4. Comparison of Iterative Receiver Performances

## 3. Parameter Learning of Trainable Functions to Maximize the Relevant Information

#### 3.1. Lookup Tables

#### 3.2. Computational Domain Technique

- Use a predefined reconstruction function $\varphi (.)$ to transfer the incoming messages ${y}_{n}$ to numbers $\varphi \left({y}_{n}\right)$ in a computational domain $\mathbb{D}$;
- Use a function $\Phi :{\mathbb{D}}^{N}\u27f6\mathcal{A}$ to process the numbers in the computational domain and to map them onto a single number $a\in \mathcal{A}$;
- Apply a scalar quantizer ${Q}_{\theta}(.)$ with ${2}^{q}-1$ ordered thresholds $\theta =[{\theta}_{0},{\theta}_{1},\dots ,{\theta}_{{2}^{q}-2}]$ on a that quantizes $a\in \mathcal{A}$ back to the set $\{0,1,\dots ,{2}^{q}-1\}$.

#### 3.3. Neural Networks

#### 3.4. Further Discussion, Other Approaches and Future Work

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Illustration of the information bottleneck method. The variables $\mathsf{X},\mathsf{Y}$ and $\mathsf{T}$ form a Markov chain $\mathsf{X}\to \mathsf{Y}\to \mathsf{T}$ and are termed the relevant, observed and compressed variables, respectively. The fundamental principle is to minimize I$(\mathsf{Y};\mathsf{T})$ while preserving I$(\mathsf{X};\mathsf{T})$.

**Figure 2.**Overview of the inputs taken and the outputs delivered by an information bottleneck algorithm.

**Figure 3.**Examples of information bottleneck graphs. The trapezoid nodes correspond to the compression mappings $p\left(t\right|y)$ and $p\left(t\right|{y}_{0},{y}_{1},{y}_{2},{y}_{3},{y}_{4})$, respectively. Both are designed to preseve I$(\mathsf{X};\mathsf{T})$.

**Figure 4.**Illustration of the information bottleneck design idea of a communication system. The digital communication receiver shall be designed such that the relevant information I$(\mathsf{X};\mathsf{T})$ is maximized from end to end.

**Figure 5.**Overview of the considered transmitter. P pilot bits are multiplexed into a length ${N}^{\mathrm{LDPC}}>>B$ LDPC codeword periodically with a distance of $B-P$ bits, and the resulting data stream is modulated using BPSK modulation.

**Figure 6.**Conventional and information bottleneck receiver chains for LDPC-encoded data transmission over a frequency-flat fading channel. The conventional receiver uses quasi-continuous received samples ${\tilde{y}}_{k}$ with double-precision and processes them in state-of-the-art detection and decoding algorithms. The information bottleneck receiver works on quantization indices and implements all signal processing using information bottleneck lookup tables.

**Figure 7.**Information bottleneck graphs of the forward and feedback channel estimation schemes for the information bottleneck receiver. The upper part of the figure shows the forward channel estimator consisting of $P-1$ two-input information bottleneck compression mappings. The lower part shows the feedback channel estimator with a similar structure to process ${N}_{\mathrm{FB}}=4$ inputs.

**Figure 8.**Information bottleneck graph of the detection scheme for the information bottleneck receiver. The detection scheme first extracts information on ${s}_{k}$ from $({y}_{k}^{\mathrm{re}},{\widehat{h}}^{\mathrm{re}})$ and $({y}_{k}^{\mathrm{im}},{\widehat{h}}^{\mathrm{im}})$ independently. Afterwards, it yields a an integer output ${t}_{k}$ from ${t}_{k}^{\mathrm{re}}$ and ${t}_{k}^{\mathrm{im}}$ that is informative about ${s}_{k}$ using the mapping $p\left({t}_{k}\right|{t}_{k}^{\mathrm{re}},{t}_{k}^{\mathrm{im}})$.

**Figure 9.**Message generation of a check node and a variable node in an iterative information bottleneck LDPC decoder. The nodes generate integer messages for their connected edges.

**Figure 10.**Bit error rates of different quantized receivers and the conventional receiver from Figure 6, which did not suffer from a quantization loss at all. The quantized information bottleneck receiver with $q=5$ bit channel output quantization, ${q}^{\mathrm{ce}}=8$ bit channel estimation and ${q}^{\mathrm{det}}=5$ bit detection and LDPC decoding met the performance of the double-precision reference receiver for ${i}_{\mathrm{FB}}=5$ feedback iterations.

**Figure 11.**Bit error rates of different receiver implementations as a function of the number ${i}_{\mathrm{FB}}$ of feedback iterations. The quantized information bottleneck receiver with $q=5$-bit channel output quantization, ${q}^{\mathrm{ce}}=8$-bit channel estimation and ${q}^{\mathrm{det}}=5$-bit detection and LDPC decoding offered performance similar to the double-precision conventional receiver for all investigated ${i}_{\mathrm{FB}}$.

**Figure 12.**General illustration of an information bottleneck signal processing unit. The signal processing unit consists of a trainable function with parameters $\theta $ that can be learned to maximize I$(\mathsf{X};\mathsf{T})$.

**Figure 14.**Bit error rates of several LDPC decoders for data transmission with BPSK over an additive white Gaussian noise channel. The applied code was a $(3,6)$ regular LDPC code. All decoders conducted ${i}_{\mathrm{max}}=50$ decoding iterations. The computational domain approach refers to [24].

**Figure 15.**Bit error rates of several LDPC decoders for data transmission with BPSK over an additive white Gaussian noise channel. The applied code was a $(3,6)$ regular LDPC code. All decoders conducted ${i}_{\mathrm{max}}=50$ decoding iterations. The computational domain approach refers to [24].

**Figure 16.**Illustration of a mapping $t={f}_{\theta}\left(\mathbf{y}\right)$ based on the nearest neighbor search. The output t is the index of the nearest neighbor ${\theta}_{t}$ of $\mathbf{y}$. It can be found by using graph-based algorithms efficiently.

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**MDPI and ACS Style**

Lewandowsky, J.; Bauch, G.; Stark, M.
Information Bottleneck Signal Processing and Learning to Maximize Relevant Information for Communication Receivers. *Entropy* **2022**, *24*, 972.
https://doi.org/10.3390/e24070972

**AMA Style**

Lewandowsky J, Bauch G, Stark M.
Information Bottleneck Signal Processing and Learning to Maximize Relevant Information for Communication Receivers. *Entropy*. 2022; 24(7):972.
https://doi.org/10.3390/e24070972

**Chicago/Turabian Style**

Lewandowsky, Jan, Gerhard Bauch, and Maximilian Stark.
2022. "Information Bottleneck Signal Processing and Learning to Maximize Relevant Information for Communication Receivers" *Entropy* 24, no. 7: 972.
https://doi.org/10.3390/e24070972