TopoCode: Topologically Informed Error Detection and Correction in Communication Systems

TL;DR

This work tackles the inadequacy of bit-level error metrics for high-data-rate, low-latency communications where semantic fidelity matters (e.g., XR and HTC). It introduces Topocode, which embeds a Persistence Diagram (PD) derived from data topology with the source data and uses the $p$-Wasserstein distance between PDs to detect and quantify message-level errors. The key contributions are (i) a minimal-redundancy error-detection mechanism based on total persistence, (ii) a PD-based, gradient-friendly error-correction scheme that minimizes the sum of $p$-Wasserstein distances across homology groups, and (iii) empirical validation on MNIST/Omniglot showing superior performance in low-SNR regimes and a compact code length. The approach is training-free, modality-agnostic, and extensible to point clouds, time-series, and text, offering a new pathway to semantic integrity in future communication systems.

Abstract

Traditional error detection and correction codes focus on bit-level fidelity, which is insufficient for emerging technologies like eXtended Reality (XR) and holographic communications requiring high-data-rate, low-latency systems. Bit-level metrics cannot comprehensively evaluate Quality-of-Service (QoS) in these scenarios. This letter proposes TopoCode which leverages Topological Data Analysis (TDA) and persistent homology to encode topological information for message-level error detection and correction. It introduces minimal redundancy while enabling effective data reconstruction, especially in low Signal-to-Noise Ratio (SNR) conditions. TopoCode offers a promising approach to meet the demands of next-generation communication systems prioritizing semantic accuracy and message-level integrity.

Figures (9)

• Figure 1: Comparison of point clouds and PDs for circle variants.
• Figure 2: Wasserstein distance between the examples in Fig. \ref{['fig:point_cloud']} with different $p$ values.
• Figure 3: Illustration of the proposed Topocode data packet structure.
• Figure 4: Illustration of topological total persistence.
• Figure 5: Example of error detection using Topocode. The PSNR and SSIM of Image-0 is 24.7 dB and 0.837, respectively. The PSNR and SSIM of Image-1 is 24.7 dB and 0.919, respectively.