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Low-Bit Quantization of Bandlimited Graph Signals via Iterative Methods

Felix Krahmer, He Lyu, Rayan Saab, Jinna Qian, Anna Veselovska, Rongrong Wang

TL;DR

This work proposes iterative noise-shaping algorithms for quantization, including sampling approaches with and without vertex replacement, that leverage the spectral properties of the graph Laplacian and exploit graph incoherence to achieve high-fidelity approximations.

Abstract

We study the quantization of real-valued bandlimited signals on graphs, focusing on low-bit representations. We propose iterative noise-shaping algorithms for quantization, including sampling approaches with and without vertex replacement. The methods leverage the spectral properties of the graph Laplacian and exploit graph incoherence to achieve high-fidelity approximations. Theoretical guarantees are provided for the random sampling method, and extensive numerical experiments on synthetic and real-world graphs illustrate the efficiency and robustness of the proposed schemes.

Low-Bit Quantization of Bandlimited Graph Signals via Iterative Methods

TL;DR

This work proposes iterative noise-shaping algorithms for quantization, including sampling approaches with and without vertex replacement, that leverage the spectral properties of the graph Laplacian and exploit graph incoherence to achieve high-fidelity approximations.

Abstract

We study the quantization of real-valued bandlimited signals on graphs, focusing on low-bit representations. We propose iterative noise-shaping algorithms for quantization, including sampling approaches with and without vertex replacement. The methods leverage the spectral properties of the graph Laplacian and exploit graph incoherence to achieve high-fidelity approximations. Theoretical guarantees are provided for the random sampling method, and extensive numerical experiments on synthetic and real-world graphs illustrate the efficiency and robustness of the proposed schemes.
Paper Structure (16 sections, 3 theorems, 61 equations, 5 figures, 3 algorithms)

This paper contains 16 sections, 3 theorems, 61 equations, 5 figures, 3 algorithms.

Table of Contents

  1. State of the art and any preliminary works
  2. Relation to Prior Work.
  3. Organization of the paper.
  4. Graph Signal Model and Preliminaries
  5. Iterative Noise-Shaping Quantization Methods
  6. Permutation-Based Iterative Quantization
  7. Random Sampling with Replacement
  8. Simulation Results
  9. Experiment 1: Effect of quantization on bandlimited graph signals.
  10. Experiment 2: Performance of the Proposed Quantization Methods Across Diverse Graphs.
  11. Experiment 3: Validation of Theoretical Error Bounds for SSS-R.
  12. Experiment 4: Quantization of Approximately Bandlimited Graph Signals on Meshes.
  13. Proofs
  14. Auxiliary Lemmas
  15. Proof of Theorem \ref{['Thm1']}
  16. ...and 1 more sections

Key Result

Theorem 3.1

Consider a bandlimited graph signal $\boldsymbol{f}$, where $\boldsymbol{f}=\bm X_r \bm \alpha$ for some $\bm \alpha\in \mathbb{R}^r$, with $c \le \|\boldsymbol{f}\|_{\infty}\le 1$. Assume that the parameter $K$ in the definition alphabet of the alphabet $\mathcal{A}_{\delta, K}$ satisfies $K\delta> where $C$ is an absolute constant. In addition, $\widetilde{\bm{q}}$ can be represented using $O\!\

Figures (5)

  • Figure 1: Performance of the Algorithm \ref{['alg:NlogN-bits']} on the bunny graphs of size ${N=2503}$ for a graph signal $\boldsymbol{f}$ with bandwidth ${r=100}$.
  • Figure 2: Illustration of performance of the proposed quantization algorithms for bandlimited graph signals on different graphs: Algorithm \ref{['alg:full-with-permuations']} with the initialization Sigma-Delta-Weight (SDW) and Step-by-Step-Serving with Replacement (SSS-R) presented in Algorithm \ref{['alg:NlogN-bits']}.
  • Figure 3: Illustration of performance of the proposed quantization algorithms for bandlimited graph signals on different graphs: Algorithm \ref{['alg:full-with-permuations']} with the initialization Sigma-Delta-Weight (SDW) and Step-by-Step-Serving with Replacement (SSS-R) presented in Algorithm \ref{['alg:NlogN-bits']}.
  • Figure 4: Illustration of theoretical and empirical error bounds of Step-by-Step-Serving with Replacement (SSS-R) algorithm presented in Algorithm \ref{['alg:NlogN-bits']}.
  • Figure 5: Illustration of performance of the proposed quantization algorithms for a general graph data on bunny graph: Memoryless quantization, Algorithm \ref{['alg:full-with-permuations']} with initialization Sigma-Delta-Weight (SDW), and Step-by-Step-Serving with Replacement (SSS-R) presented in Algorithm \ref{['alg:NlogN-bits']}.

Theorems & Definitions (6)

  • Definition 2.1
  • Theorem 3.1
  • Lemma 5.1
  • proof
  • Lemma 5.2
  • proof