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Reliable one-bit quantization of bandlimited graph data via single-shot noise shaping

Johannes Maly, Anna Veselovska

TL;DR

The paper tackles quantizing bandlimited graph signals with very few bits while preserving low-pass information. It introduces single-shot noise shaping (SSNS), combining a preprocessing step with memoryless scalar quantization to encode an $N$-dimensional graph signal into an $N$-dimensional quantized representation that supports arbitrary bit-depth, including 1-bit. A key theoretical result bounds the quantization error after low-pass filtering by $ rac{ ext{QÉ}_{oldsymbol{L}_r}(oldsymbol{f},oldsymbol{q})}{ orm{oldsymbol{f}}_2} \,ig\le C \, 2^{-B} \, mu(oldsymbol{X}_r) \, rac{r}{\, oot 2 t N}$, showing exponential decay in $B$ with a graph-incoherence factor. The work contrasts SSNS with iterative noise-shaping methods, demonstrating tighter worst-case scaling under comparable bit budgets, and validates the approach across diverse graph topologies and a 3D halftoning task, highlighting practical applicability for scalable graph learning and compression.

Abstract

Graph data are ubiquitous in natural sciences and machine learning. In this paper, we consider the problem of quantizing graph structured, bandlimited data to few bits per entry while preserving its information under low-pass filtering. We propose an efficient single-shot noise shaping method that achieves state-of-the-art performance and comes with rigorous error bounds. In contrast to existing methods it allows reliable quantization to arbitrary bit-levels including the extreme case of using a single bit per data coefficient.

Reliable one-bit quantization of bandlimited graph data via single-shot noise shaping

TL;DR

The paper tackles quantizing bandlimited graph signals with very few bits while preserving low-pass information. It introduces single-shot noise shaping (SSNS), combining a preprocessing step with memoryless scalar quantization to encode an -dimensional graph signal into an -dimensional quantized representation that supports arbitrary bit-depth, including 1-bit. A key theoretical result bounds the quantization error after low-pass filtering by , showing exponential decay in with a graph-incoherence factor. The work contrasts SSNS with iterative noise-shaping methods, demonstrating tighter worst-case scaling under comparable bit budgets, and validates the approach across diverse graph topologies and a 3D halftoning task, highlighting practical applicability for scalable graph learning and compression.

Abstract

Graph data are ubiquitous in natural sciences and machine learning. In this paper, we consider the problem of quantizing graph structured, bandlimited data to few bits per entry while preserving its information under low-pass filtering. We propose an efficient single-shot noise shaping method that achieves state-of-the-art performance and comes with rigorous error bounds. In contrast to existing methods it allows reliable quantization to arbitrary bit-levels including the extreme case of using a single bit per data coefficient.
Paper Structure (9 sections, 2 theorems, 17 equations, 6 figures, 3 algorithms)

This paper contains 9 sections, 2 theorems, 17 equations, 6 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Contribution
  3. Notation
  4. Preliminaries on graph data
  5. Quantizing graph data
  6. Single-shot noise-shaping (SSNS)
  7. Comparison with iterative noise-shaping methods
  8. Numerical Experiments
  9. Discussion

Key Result

Theorem 3.1

There exists an absolute constant $C > 0$ such that the following holds. Let $\mathcal{G} = (\mathcal{V},\mathcal{E},\boldsymbol{W})$ be an undirected graph. Then, for any $r$-bandlimited $\boldsymbol{f} \in \mathbb{R}^N$ with $\| \boldsymbol{f} \|_\infty = 1$, we have that $\boldsymbol{q}$ computed where $\boldsymbol{L}_r = \boldsymbol{X}_r\boldsymbol{X}_r^T$ is the brick-wall filter of bandwidth

Figures (6)

  • Figure 1: Illustration of Algorithm \ref{['alg:Preprocessing']} in $\mathbb{R}^3$. The algorithm walks in a kernel direction (green plane) until it hits the boundary of the scaled $\ell_\infty$ ball $cB_\infty$ such that at least one entry of $\boldsymbol{z}_k$ is set to $\pm c$. After reducing the feasible kernel directions (dashed green line), the algorithm repeats the procedure until at most $r$ entries of $\boldsymbol{z}_k$ are not equal to $\pm c$.
  • Figure 2: Performance of the proposed SSNS quantization algorithm on different graph structures. For each graph (left), the corresponding semilog plot of the relative reconstruction error (right) is shown for different quantization bit budgets $B = 1,2,4$.
  • Figure 3: Comparison of the Bunny data and this quantized version: (a) original, (b–d) quantized versions, (e) data frequency spectrum, and (f–h) quantization errors. Each column corresponds to a specific method.
  • Figure 4: Comparison of the performance of SSNS to the theoretical worst-case guarantees \ref{['thm:ErrorBound']} and to the Step-by-Step-Serving with Replacement (SSS-R) method.
  • Figure 5: Average relative error versus bit-depth for SSNS across different graph topologies, compared with the theoretical bound proportional to $2^{-B}$. Results are averaged over 50 random signals with bandwidth $r=200$.
  • ...and 1 more figures

Theorems & Definitions (4)

  • Definition 2.1
  • Theorem 3.1
  • Theorem 3.2: maly2023simple
  • proof : Proof of Theorem \ref{['thm:ErrorBound']}