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

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.

💡 Research Summary

Title: Reliable one‑bit quantization of bandlimited graph data via single‑shot noise shaping

Problem Setting
Graph‑structured data appear in many scientific and machine learning contexts. Often such data are smooth on the graph, i.e., their Graph Fourier Transform (GFT) is concentrated on the low‑frequency eigenvectors of the normalized Laplacian. Quantizing these signals with very few bits per entry while preserving the information needed for low‑pass filtering is crucial for communication‑efficient distributed processing, memory‑constrained storage, and efficient training of Graph Neural Networks (GNNs). Existing approaches either rely on ΣΔ‑type iterative noise shaping (Krahmer et al., 2023, 2026) which needs a logarithmic overhead in the alphabet size, or on over‑complete graph Fourier frames (Reingruber & Matz, 2025) that increase dimensionality. Both are computationally heavier and do not support true one‑bit quantization.

Key Contribution
The authors introduce a Single‑Shot Noise Shaping (SSNS) scheme that quantizes an r‑bandlimited graph signal f ∈ ℝⁿ using a single preprocessing step followed by a standard memoryless scalar quantizer (MSQ). The preprocessing (Algorithm 1) is a deterministic walk in the kernel of the low‑frequency eigenvector matrix X_rᵀ. Starting from z₀ = f and a radius c = ‖f‖∞, the algorithm repeatedly adds a kernel vector b until the ℓ∞‑norm hits the boundary c. After at most r iterations, the resulting vector b_f satisfies:

• X_rᵀ b_f = X_rᵀ f (the low‑frequency content is unchanged);
• All but at most r entries of b_f are exactly ±c (here c = 1 after normalization).

Thus, when a mid‑rise alphabet A_B = {−1, −1+2·2⁻ᴮ, …, 1} is applied, the quantizer Q produces q = Q(b_f) with zero quantization error on the saturated entries; only the at‑most‑r unsaturated entries contribute to the error.

Theoretical Guarantees
Theorem 3.1 shows that for any r‑bandlimited signal, \