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Joint Coding-Modulation for Digital Semantic Communications via Variational Autoencoder

Yufei Bo, Yiheng Duan, Shuo Shao, Meixia Tao

arXiv:2310.06690v2cs.ITeess.SPmath.IT

TL;DR

Digital semantic communication faces the challenge of mapping neural-network outputs to discrete symbols while accounting for channel conditions. This paper proposes a VAE-based joint coding-modulation framework that learns symbol transition probabilities and trains coding and modulation together. Experiments report advantages over quantization-based digital methods across channel conditions, rates, and modulation orders, with a shrinking performance gap to analog communication at higher orders.

  • Problem

    Neural semantic encoders commonly transmit continuous outputs, while digital modulation is non-differentiable and existing digital methods may not match modulation to channel conditions.

    Source: §I · pp. 1–2

  • Method

    The VAE-based JCM framework learns transition probabilities from source data to discrete constellation symbols and jointly trains coding, modulation, and decoding with channel noise considered.

    Source: §I · p. 2

  • Results

    JCM outperforms quantization-based digital methods across channel conditions, transmission rates, and modulation orders, and its performance gap to analog semantic communication decreases as modulation order increases.

    Source: §I · p. 2, §V · p. 11

  • Takeaways & Limitations

    The paper concludes that using the highest modulation order supported by a transmission device is a convenient way to conduct digital modulation in this framework.

    Source: §V · p. 11

Abstract

from arXiv · show

Semantic communications have emerged as a new paradigm for improving communication efficiency by transmitting the semantic information of a source message that is most relevant to a desired task at the receiver. Most existing approaches typically utilize neural networks (NNs) to design end-to-end semantic communication systems, where NN-based semantic encoders output continuously distributed signals to be sent directly to the channel in an analog fashion. In this work, we propose a joint coding-modulation (JCM) framework for digital semantic communications by using variational autoencoder (VAE). Our approach learns the transition probability from source data to discrete constellation symbols, thereby avoiding the non-differentiability problem of digital modulation. Meanwhile, by jointly designing the coding and modulation process together, we can match the obtained modulation strategy with the operating channel condition. We also derive a matching loss function with information-theoretic meaning for end-to-end training. Experiments on image semantic communication validate the superiority of our proposed JCM framework over the state-of-the-art quantization-based digital semantic coding-modulation methods across a wide range of channel conditions, transmission rates, and modulation orders. Furthermore, its performance gap to analog semantic communication reduces as the modulation order increases while enjoying the hardware implementation convenience.

I. INTRODUCTION

The paper proposes an end-to-end VAE-based joint coding-modulation framework to make digital semantic communication differentiable and channel-aware. Its system transmits discrete constellation symbols and uses parallel decoders to recover both source data and task-dependent semantic information.

I. INTRODUCTION

  • Digital semantic communication remains challenging because discrete modulation is non-differentiable for gradient-based training, while existing quantization methods may fail to match channel conditions.The paper also notes that hard-decision joint quantization can lose information.Source: §I · pp. 1–2
  • JCM learns a transition probability from source data to constellation symbols and jointly trains coding, modulation, and decoding with channel noise considered.This probabilistic design addresses digital modulation’s non-differentiability and aims to match modulation to channel conditions.Source: §I · p. 2

A. System Model

  • The system models source data X and its task-dependent semantic information S as inputs to be recovered after transmission through a noisy channel.The transmitter and receiver share a dataset of source samples and their corresponding semantic information.Source: A. System Model · pp. 2–3
  • A probabilistic encoder-modulator learns symbol probabilities and samples a discrete constellation sequence, which is scaled to meet an average transmit-power constraint.The system considers M-order modulation and an AWGN channel characterized by SNR.Source: A. System Model · p. 3
  • Two parallel probabilistic decoders estimate the source data and semantic information directly from the received sequence.The parallel design avoids the information loss that the paper says would arise from inferring semantics only after reconstructing the source.Source: A. System Model · p. 3

B. Objective Function

The objective is to preserve information about both the source and its semantics in the received sequence. Because the mutual-information objective is difficult to estimate and omits decoder parameters, the paper motivates a jointly trainable lower-bound loss.

B. Objective Function

  • The JCM objective seeks to maximize mutual information between the received sequence and both the source data and semantic information.The paper defines it as the sum of two mutual-information terms, unlike the subtraction used by the cited information-bottleneck formulation.Source: B. Objective Function · p. 3
  • Directly optimizing the mutual-information objective is difficult because the high-dimensional variables have complicated distributions, and the objective includes only encoder-modulator parameters.Joint training must also account for both receiver decoders.Source: B. Objective Function · p. 4

III. VARIATIONAL LEARNING OF THE JCM FRAMEWORK

  • The paper proposes a variational-inference lower bound on the objective that jointly optimizes the encoder-modulator and both decoders.The bound is intended to make the informative constellation sequence decodable by the receiver.Source: B. Objective Function · p. 4
  • Training targets the transmitter’s transition probabilities and the receiver’s source and semantic posterior probabilities.The paper describes updating the decoders under a fixed transmitter distribution, then updating the transmitter based on decoder performance.Source: §III · p. 4

A. Loss Function Design Based on Variational Inference Lower Bound

The paper derives a tractable variational lower bound for joint training and structures the encoder-modulator’s transition probabilities to reduce their learning burden. The model assumes conditionally independent symbols and gives specialized forms for common modulation schemes.

A. Loss Function Design Based on Variational Inference Lower Bound

  • The training procedure alternates between updating receiver posteriors under a fixed transmitter distribution and updating the transmitter based on decoder performance.The main loss-design challenge is to make the decoder posteriors approach the true posteriors.Source: A. Loss Function Design Based on… · p. 4
  • Variational inference yields a tractable lower bound on the mutual-information objective, using approximate decoder posteriors to measure distance from the true posteriors.Maximizing the bound trains the encoder-modulator and decoders; the paper says it can be computed from empirical data and estimated expectations.Source: A. Loss Function Design Based on… · p. 4

B. Transition Probability Model of The Encoder-Modulator

  • Assuming conditional independence across symbols reduces the learned transition-probability categories from M^n to M·n for an n-symbol sequence.The neural network outputs a categorical distribution for each constellation symbol.Source: B. Transition Probability Model of The… · p. 5
  • For rectangular M-QAM, the I and Q channels are treated as conditionally independent, requiring two M-category distributions per symbol instead of one M-category distribution.For BPSK, the transition probability reduces to a Bernoulli distribution.Source: B. Transition Probability Model of The… · p. 5
  • The transition-probability loss can be generalized to arbitrary constellation maps, including non-uniform maps.Source: B. Transition Probability Model of The… · p. 5

C. Loss Function for Image Semantic Communications

The section specializes the variational information lower bound (VILB) for image semantic communication by applying image and label assumptions. The resulting loss combines classification and image-recovery terms and supports joint optimization of the model parameters.

C. Loss Function for Image Semantic Communications

  • The image-semantic loss (13) is derived from VILB using image-source assumptions and empirical distributions.The derivation models source and reconstructed images as factorized Gaussians and uses classification labels as discrete semantic information.Source: C. Loss Function for Image Semantic… · p. 5
  • The loss is a weighted sum of cross entropy for semantic classification and mean squared error for source-image recovery.Cross entropy measures the distance between the true and approximate semantic posterior; MSE compares the source and recovered image.Source: C. Loss Function for Image Semantic… · p. 5
  • Replacing probability distributions with empirical distributions yields a loss that jointly optimizes encoder-modulator and decoder parameters θ, ϕ, and ψ.The classification-label term reduces to cross entropy, while the Gaussian image term reduces to MSE.Source: C. Loss Function for Image Semantic… · p. 5

D. Differentiable Constellation Symbol Generation

The section addresses gradient estimation for discrete constellation symbols by using Gumbel-Softmax reparameterization. It samples symbols in the forward pass and substitutes a differentiable softmax approximation for argmax during backpropagation.

D. Differentiable Constellation Symbol Generation

  • Because θ determines the distribution inside the expectation, its gradient cannot be estimated by directly applying the Monte Carlo estimator.The score-function identity rewrites the gradient as an expectation involving ∇θ log p_en(z|x, θ).Source: D. Differentiable Constellation Symbol… · pp. 5–6
  • The score-function approach permits Monte Carlo gradient estimation, but it can have high variance and slow convergence, increasing training computation when variance-reduction methods are used.The passage notes that variance-reduction techniques are often used alongside this estimator.Source: D. Differentiable Constellation Symbol… · p. 6
  • Gumbel-Softmax samples categorical constellation symbols with Gumbel-Max in the forward pass and replaces argmax with softmax for backpropagation.The softmax approximation is controlled by temperature ρ, and makes the loss differentiable for estimating gradients with respect to θ.Source: D. Differentiable Constellation Symbol… · pp. 6–7

IV. EXPERIMENT RESULTS

The experiments are conducted using a CPU and GPU with a PyTorch setup.

IV. EXPERIMENT RESULTS

  • Experiments use an Intel Xeon Silver 4214R CPU and an Nvidia GeForce RTX 3090 Ti GPU with PyTorch and CUDA 11.4.The GPU has 24 GB of memory.Source: §IV · p. 7

A. Experiment Settings

The experiment settings specify the image dataset and semantic label, model architecture and training hyperparameters, and benchmark methods and evaluation criteria.

1) Datasets:

  • CIFAR10 is the default dataset, with its classification label serving as semantic information.The dataset contains 60,000 32 × 32 color images across 10 classes, split into 50,000 training and 10,000 test images.Source: 1) Datasets · p. 7

2) Neural Network Architecture And Hyper-parameters:

  • The proposed model uses a Resnet probabilistic encoder-modulator and Spinalnet semantic decoder, with Resnet and depth-to-space for source reconstruction.Training uses batch size 32, Adam, and cosine annealing over 300 epochs; λ varies with system parameters, and Table II lists its values for varying SNR and modulation at n = 128.Source: 2) Neural Network Architecture And… · p. 7, Table II · p. 7, Table I · p. 7

3) Benchmarks:

  • The JCM framework is compared with four benchmarks—Analog, DeepJSCC-Q, Uniform, and NN quantization—and evaluated by PSNR for image recovery and accuracy for image classification.Source: 3) Benchmarks · pp. 7–8

B. Performances against Varying SNRs

JCM outperforms quantization-based digital baselines across SNR and modulation settings, while its performance gap to analog narrows as modulation order increases. Rate sweeps and recovered-image examples further characterize its performance.

B. Performances against Varying SNRs

C. Performances against Varying Transmission Rates

Raw JCM

  • Fig. 5 presents recovered-image examples at 0 dB with 1024 channel uses and 16QAM.Source: Figure 5 · p. 10

D. Performances against Different Modulation Orders

The JCM framework performs better than separate coding-modulation designs across varied conditions, and its performance improves with modulation order. The conclusion presents jointly learned VAE-based coding and modulation as a way to match channel conditions and reports a shrinking gap to analog communication at higher orders.

D. Performances against Different Modulation Orders

V. CONCLUSION

  • The VAE-based JCM framework jointly learns coding and modulation to match channel conditions and uses a variational-inference loss for source reconstruction and semantic-task execution.The probabilistic model addresses the non-differentiability of digital modulation.Source: §V · p. 11
  • JCM outperforms separate coding-modulation design across channel conditions, transmission rates, and modulation orders; its gap to analog communication narrows as modulation order increases.The conclusion recommends using the highest modulation order the device supports.Source: §V · p. 11

APPENDIX A PROOF OF THEOREM 1

The proof uses variational inference to lower-bound MI-OBJ by bringing decoder probabilities toward true posterior distributions. It then applies the bound using the encoder-modulator transition probabilities.

APPENDIX A PROOF OF THEOREM 1

  • Variational inference supplies a lower bound on MI-OBJ by approximating the true posterior distributions with NN-parameterized decoder probabilities.The proof uses nonnegativity of KL divergence to establish the lower bound for the semantic-information term.Source: Appendix A · pp. 12–13
  • The Markov chain S → X → Z → Ẑ → (Ŝ, X̂) lets the proof express the received-sequence probability through the encoder-modulator transition probability.The proof then introduces this probability into Lemma 1.Source: Appendix A · p. 13
  • The resulting variational-inference lower bound is parameterized by encoder parameters θ and decoder parameters ϕ and ψ.Source: Appendix A · p. 13

APPENDIX B PROOF OF PROPOSITION 1

The appendix gives transition-probability formulations for BPSK and rectangular M-QAM. For QAM, it treats the I and Q channels as conditionally independent and substitutes the resulting expression into the earlier model.

APPENDIX B PROOF OF PROPOSITION 1

  • For BPSK, the two constellation symbols are 1 and −1, so the categorical distribution of each Z_i reduces to a Bernoulli distribution.The transition probability is written using q_i|x,θ and its complement.Source: Appendix B · p. 13

APPENDIX C PROOF OF PROPOSITION 2

  • For rectangular M-QAM, the proof assumes conditionally independent I and Q channels and expresses each constellation symbol through its component amplitudes.Source: Appendix C · p. 13
  • The QAM symbol probability is expressed using the I- and Q-channel probabilities, and substituting the resulting expressions into equation (8) yields equation (10).Source: Appendix C · p. 13

APPENDIX D PROOF OF COROLLARY 1

For image semantic communication, the proof uses a variational approximation to the source posterior under Gaussian assumptions and treats semantic labels as categorical. Replacing distributions with empirical distributions yields the image-communication loss.

APPENDIX D PROOF OF COROLLARY 1

  • Under the source-data Gaussian assumptions, the decoder’s variational distribution approximates the true posterior p(x|ẑ, ψ).The derivation also uses xᵀAx = tr(Axxᵀ).Source: Appendix D · p. 13
  • Image classification labels are treated as discrete semantic information with a categorical distribution.Source: Appendix D · p. 13
  • Replacing probability distributions with empirical distributions and substituting the resulting expressions into VILB (4) gives the image semantic-communication loss.Source: Appendix D · pp. 13–14
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