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A Copula-based Semantics-Structure Minimization Framework for QoS Guaranteed Wireless Communications

Xinke Jian, Zhiyuan Ren, Wenchi Cheng

TL;DR

This work establishes a rigorous axiomatic foundation for semantic QoS in wireless communications by showing that the minimal structural semantics of an image are captured by the family of pairwise rank-Copulas and by introducing the D_pc distortion metric based on Jensen–Shannon divergence. It derives theoretical guarantees, including sample complexity, rate–distortion bounds, end-to-end SLA reliability, and a semantic source–channel separation theorem, which together enable predictable semantic system design. The authors validate the framework with decoupled experiments, demonstrating that D_pc is both task-relevant and axiom-compliant, while traditional perceptual metrics may fail to align with semantic fidelity. The resulting Rate–Computation–Reliability map provides a practical design tool to trade bandwidth for computation to meet specified semantic QoS targets, pushing semantic communications toward a principled engineering science.

Abstract

Current empirically driven research on semantic communication lacks a unified theoretical foundation, preventing quantifiable Quality of Service guarantees, particularly for transmitting minimal structural semantics in emergency scenarios. This deficiency limits its evolution into a predictable engineering science. To address this, we establish a complete theoretical axiomatic basis for this problem. We propose four axioms and rigorously prove that the family of pairwise rank-Copulas is the minimal sufficient representation for minimal structural semantics. Based on this, we construct a semantic distortion metric, centered on the Jensen-Shannon divergence. We then establish the core theoretical boundaries of the framework: sample complexity bounds; rate-distortion bounds; an end-to-end Service Level Agreements theorem; and a semantic source-channel separation theorem, which provides a provable Quality of Service guarantee. Finally, we validate our framework through decoupled experiments, empirically demonstrating that our core metric strictly adheres to our foundational axioms while standard perceptual metrics fail to do so.

A Copula-based Semantics-Structure Minimization Framework for QoS Guaranteed Wireless Communications

TL;DR

This work establishes a rigorous axiomatic foundation for semantic QoS in wireless communications by showing that the minimal structural semantics of an image are captured by the family of pairwise rank-Copulas and by introducing the D_pc distortion metric based on Jensen–Shannon divergence. It derives theoretical guarantees, including sample complexity, rate–distortion bounds, end-to-end SLA reliability, and a semantic source–channel separation theorem, which together enable predictable semantic system design. The authors validate the framework with decoupled experiments, demonstrating that D_pc is both task-relevant and axiom-compliant, while traditional perceptual metrics may fail to align with semantic fidelity. The resulting Rate–Computation–Reliability map provides a practical design tool to trade bandwidth for computation to meet specified semantic QoS targets, pushing semantic communications toward a principled engineering science.

Abstract

Current empirically driven research on semantic communication lacks a unified theoretical foundation, preventing quantifiable Quality of Service guarantees, particularly for transmitting minimal structural semantics in emergency scenarios. This deficiency limits its evolution into a predictable engineering science. To address this, we establish a complete theoretical axiomatic basis for this problem. We propose four axioms and rigorously prove that the family of pairwise rank-Copulas is the minimal sufficient representation for minimal structural semantics. Based on this, we construct a semantic distortion metric, centered on the Jensen-Shannon divergence. We then establish the core theoretical boundaries of the framework: sample complexity bounds; rate-distortion bounds; an end-to-end Service Level Agreements theorem; and a semantic source-channel separation theorem, which provides a provable Quality of Service guarantee. Finally, we validate our framework through decoupled experiments, empirically demonstrating that our core metric strictly adheres to our foundational axioms while standard perceptual metrics fail to do so.

Paper Structure

This paper contains 26 sections, 13 theorems, 19 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Axiomatic Foundation and Semantic Representation
  3. The Axiomatic System
  4. Derivation of Representation and Metric
  5. Theoretical boundary Guarantees
  6. Sample Complexity of Semantic Representation
  7. Rate-Distortion Bounds for Semantic Compression
  8. End-to-End SLA Reachability
  9. Semantic Source-Channel Separation
  10. Performance Evaluation
  11. Experimental Setup
  12. Validation of $D_{\mathrm{pc}}$ as a Task-Oriented Metric (The "Ruler")
  13. Validation of Foundational Axioms (The "Soundness")
  14. Validation of Theoretical Bounds (The "Map")
  15. Rate-Distortion of Semantic Representation (Validating Thm. 3)
  16. ...and 11 more sections

Key Result

Theorem 1

In the model class defined by Axioms A1 (Monotone Invariance) and A2 (Pairwise Stationarity), the family of pairwise rank-Copulas $\{C_\delta(I)\}_{\delta \in \Delta}$ is the minimal sufficient statistic for the dependency structure of the image $I$.

Figures (5)

  • Figure 1: Illustration of the derivation from an image to its Copula representation. (a) An original image $I$ and its monotonically transformed version $T(I)$ have different perceptual appearances but identical structural semantics. (b) The Rank Transform, enforced by Axiom A1 , maps both images to the same unique rank image $U$. (c) For a given displacement $\delta$, pairwise sampling extracts the rank relationships $(U(p), U(p+\delta))$ . These pairs are collected into a 2D histogram, which forms the empirical Copula $C_\delta$—our "relationship fingerprint".
  • Figure 2: Validation of Axiom A1 (Monotone Invariance). (a) Original $I$, (b) grayscale $T_1(I)$, and (c) high-contrast $T_2(I)$ are semantically equivalent under A1 despite perceptual differences. (d) All three produce the identical empirical Copula $C_{\delta}$ (for $\delta=(1,0)$), proving the Copula isolates the dependency structure from marginal statistics
  • Figure 3: Correlation between AI task performance (mAP) and distortion metrics across 15 degradation levels (JPEG, Blur, Noise). The SRCC ($\rho$) values show that both $D_{\mathrm{pc}}$ (Semantic) and LPIPS (Perceptual) are strong predictors of mAP degradation, while PSNR (Pixel) is significantly less correlated.
  • Figure 4: Empirical validation of the rate-distortion (R-D) performance of the Copula semantic representation (data from Sec. \ref{['sec:evaluation']}.C.1). This curve confirms the compressibility of semantics and provides the $g(R)$ function for our SLA theory (Thm. \ref{['thm:rd_bounds']}).
  • Figure 5: Visualization of the Rate-Computation-Reliability trade-off surface derived from our SLA theorem (Thm. \ref{['thm:sla']}) and empirical data from Fig. \ref{['fig:rd_curve']}. This "map" allows engineers to quantitatively trade bandwidth (Rate) for computation (Computation) to meet a specific semantic reliability (Error) target.

Theorems & Definitions (22)

  • Theorem 1: Minimal Sufficient Representation of Minimal Structural Semantics
  • Definition 1: Pairwise Rank-Copula Semantic Representation
  • Definition 2: Pairwise Copula Consistency ($D_{\mathrm{pc}}$)
  • Theorem 2: Empirical Copula Concentration Bound
  • Theorem 3: Semantic Rate-Distortion Bounds
  • Theorem 4: $(1-\delta_{\mathrm{SLA}})$-SLA Reachability
  • Theorem 5: Semantic Source-Channel Separation
  • Theorem 6: Minimal Sufficient Representation of Structural Semantics
  • proof
  • Theorem 7: Empirical Copula Concentration
  • ...and 12 more