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Error Feedback Approach for Quantization Noise Reduction of Distributed Graph Filters

Xue Xian Zheng, Tareq Al-Naffouri

TL;DR

The paper addresses quantization noise in distributed graph filters by introducing a quantization error feedback mechanism built on a diagonal feedback matrix $\mathbf{D}$, extending error spectrum shaping to the graph domain. It derives closed-form noise-weight expressions for both FIR and ARMA graph filters under deterministic and stochastic topologies, supported by Lyapunov-based and gradient-based analyses. Theoretical results yield explicit coefficients $\alpha_{i,\cdot}$ that minimize noise amplification, and kernel-based methods enable efficient computation of expectations in the stochastic setting. Numerical experiments on a 64-node David sensor network show substantial SNR gains (up to several dB) across all filter types, demonstrating robustness to topology variation and outperforming non-feedback baselines. Overall, the work provides a practical, theory-backed approach to reliable distributed graph filtering with finite-bit communication.

Abstract

This work introduces an error feedback approach for reducing quantization noise of distributed graph filters. It comes from error spectrum shaping techniques from state-space digital filters, and therefore establishes connections between quantized filtering processes over different domains. Quantization noise expression incorporating error feedback for finite impulse response (FIR) and autoregressive moving average (ARMA) graph filters are both derived with regard to time-invariant and time-varying graph topologies. Theoretical analysis is provided, and closed-form error weight coefficients are found. Numerical experiments demonstrate the effectiveness of the proposed method in noise reduction for the graph filters regardless of the deterministic and random graph topologies.

Error Feedback Approach for Quantization Noise Reduction of Distributed Graph Filters

TL;DR

The paper addresses quantization noise in distributed graph filters by introducing a quantization error feedback mechanism built on a diagonal feedback matrix , extending error spectrum shaping to the graph domain. It derives closed-form noise-weight expressions for both FIR and ARMA graph filters under deterministic and stochastic topologies, supported by Lyapunov-based and gradient-based analyses. Theoretical results yield explicit coefficients that minimize noise amplification, and kernel-based methods enable efficient computation of expectations in the stochastic setting. Numerical experiments on a 64-node David sensor network show substantial SNR gains (up to several dB) across all filter types, demonstrating robustness to topology variation and outperforming non-feedback baselines. Overall, the work provides a practical, theory-backed approach to reliable distributed graph filtering with finite-bit communication.

Abstract

This work introduces an error feedback approach for reducing quantization noise of distributed graph filters. It comes from error spectrum shaping techniques from state-space digital filters, and therefore establishes connections between quantized filtering processes over different domains. Quantization noise expression incorporating error feedback for finite impulse response (FIR) and autoregressive moving average (ARMA) graph filters are both derived with regard to time-invariant and time-varying graph topologies. Theoretical analysis is provided, and closed-form error weight coefficients are found. Numerical experiments demonstrate the effectiveness of the proposed method in noise reduction for the graph filters regardless of the deterministic and random graph topologies.

Paper Structure

This paper contains 13 sections, 9 theorems, 23 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Deterministic graph filters
  4. Stochastic graph filters
  5. Quantization model
  6. Quantization Error Feedback
  7. Error feedback on deterministic FIR graph filter
  8. Error feedback on deterministic ARMA graph filter
  9. Error feedback on stochastic FIR graph filter
  10. Error feedback on stochastic ARMA graph filter
  11. Numerical Experiments
  12. Conclusion
  13. APPENDIX I

Key Result

Proposition 1

For feedbacked quantization noise output $\widetilde{\mathbf{y}}_{\mathbf{n}_{k-1}} = \mathbf{H}_{\mathbf{n}_{k-1}} (\mathbf{S})(\mathbf{S}-\mathbf{D})\mathbf{n}_{k-1}$ coming from $\mathbf{n}_{k-1}$ with sub graph filter $\mathbf{H}_{\mathbf{n}_{k-1}} (\mathbf{S}) = \sum_{\kappa =k}^{K}\phi_{\kapp where $\mathbf{G}_{\mathbf{n}_{k-1}}(\mathbf{S})=\mathbf{H}^{T}_{\mathbf{n}_{k-1}} (\mathbf{S})\mat

Figures (2)

  • Figure 1: An illustration of quantization error feedback on deterministic FIR graph filter. Let $\mathbf{D}$ be diagonal, it is interpreted that each node locally stores its quantization error, applies a weight factor, and then integrates it into the fusion process.
  • Figure 2: Performance comparison.(a) The quantization bits for these filters are $b = \{14\times 1, 16\times 2, 20\times 2, 25\times 2\}$. (b) ARMA$_{1}$ with $b ={6}$. (c) FIR of order $K=6$ with $b ={12}$. (d) ARMA$_{1}$ with $b ={4}$ under $p=0.95$.

Theorems & Definitions (14)

  • Proposition 1
  • Theorem 1
  • proof
  • Proposition 2
  • Theorem 2
  • proof
  • Proposition 3
  • Theorem 3
  • proof
  • Proposition 4
  • ...and 4 more