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arxiv: 2506.01404 · v2 · submitted 2025-06-02 · 💻 cs.LG · cs.MA· cs.SY· eess.SY

Quantitative Error Feedback for Quantization Noise Reduction of Filtering over Graphs

Xue Xian Zheng , Weihang Liu , Xin Lou , Stefan Vlaski , Tareq Al-Naffouri This is my paper

Pith reviewed 2026-05-19 11:15 UTC · model grok-4.3

classification 💻 cs.LG cs.MAcs.SYeess.SY
keywords quantization noiseerror feedbackgraph filteringdistributed algorithmsquantized communicationdecentralized optimizationrandom graphsasynchronous updates
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The pith

Quantitative error feedback compensates quantization noise exactly in distributed graph filtering by feeding back isolated errors with closed-form optimal coefficients.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a framework that isolates quantization noise during message passing in graph filters and feeds it back quantitatively for compensation, drawing from error spectrum shaping in digital filters. It analyzes this under deterministic graphs, random graphs, and random asynchronous node updates, deriving closed-form solutions for the best feedback coefficients in each case. The result is lower overall error in the filtering output without extra communication overhead. This mechanism also integrates into broader decentralized optimization routines to push down their error floors. Experiments confirm consistent gains over standard quantization approaches in accuracy and stability.

Core claim

By quantitatively feeding back the exact quantization noise term within the graph filtering update rules, the framework achieves substantial reduction in quantization-induced error, with explicit closed-form expressions for the optimal feedback coefficients that apply to deterministic filtering, filtering on random graphs, and filtering under random node-asynchronous updates.

What carries the argument

The quantitative error feedback mechanism that isolates the quantization noise term and subtracts a scaled version of it from subsequent updates to achieve exact compensation.

If this is right

  • Quantization noise effect is significantly reduced in distributed graph filtering across the three analyzed scenarios.
  • Closed-form optimal error feedback coefficients are available for deterministic, random-graph, and asynchronous cases.
  • The same mechanism integrates directly into communication-efficient decentralized optimization to achieve lower error floors.
  • Numerical results show consistent outperformance versus conventional quantization strategies in both accuracy and robustness.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The approach may extend to other message-passing algorithms such as consensus or distributed learning where quantization is a bottleneck.
  • In sensor networks or edge devices with strict bit constraints, this could enable higher effective precision without increasing bandwidth.
  • Similar feedback ideas might apply to quantization in neural network inference on graphs or other structured data.

Load-bearing premise

The quantization noise term can be precisely isolated from the signal and fed back for exact compensation without needing extra communication bits or breaking the rules of the three filtering scenarios.

What would settle it

An experiment on a small deterministic graph where the measured steady-state error after applying the derived optimal coefficients fails to match the predicted reduction or requires additional bits to isolate the noise would falsify the central claim.

Figures

Figures reproduced from arXiv: 2506.01404 by Stefan Vlaski, Tareq Al-Naffouri, Weihang Liu, Xin Lou, Xue Xian Zheng.

Figure 1
Figure 1. Figure 1: MSD versus filter order/iteration of quantized graph filtering on deterministic processes. [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: MSD versus probability/iteration of quantized graph filtering on random processes. [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: MSD versus iteration of quantized regression over a network, [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
read the original abstract

This paper introduces an innovative error feedback framework designed to mitigate quantization noise in distributed graph filtering, where communications are constrained to quantized messages. It comes from error spectrum shaping techniques from state-space digital filters, and therefore establishes connections between quantized filtering processes over different domains. In contrast to existing error compensation methods, our framework quantitatively feeds back the quantization noise for exact compensation. We examine the framework under three key scenarios: (i) deterministic graph filtering, (ii) graph filtering over random graphs, and (iii) graph filtering with random node-asynchronous updates. Rigorous theoretical analysis demonstrates that the proposed framework significantly reduces the effect of quantization noise, and we provide closed-form solutions for the optimal error feedback coefficients. Moreover, this quantitative error feedback mechanism can be seamlessly integrated into communication-efficient decentralized optimization frameworks, enabling lower error floors. Numerical experiments validate the theoretical results, consistently showing that our method outperforms conventional quantization strategies in terms of both accuracy and robustness.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper introduces a quantitative error feedback framework, adapted from error spectrum shaping in digital filters, to mitigate quantization noise in distributed graph filtering under quantized communications. It derives closed-form optimal error feedback coefficients for exact compensation in three scenarios—deterministic graph filtering, filtering over random graphs, and filtering with random node-asynchronous updates—claims significant noise reduction via rigorous analysis, and shows integration into decentralized optimization with lower error floors, supported by numerical experiments.

Significance. If the exact local compensation holds without hidden communication or state requirements, the closed-form solutions and DSP-to-graph-filtering connections could meaningfully advance communication-efficient distributed algorithms on graphs, particularly in asynchronous or random settings common to sensor networks and decentralized ML. The numerical validation showing consistent outperformance over standard quantization is a concrete strength.

major comments (2)
  1. [§4.3] §4.3 (random node-asynchronous updates): the derivation of optimal feedback coefficients assumes each node can locally isolate the precise quantization error e = x − Q(x) and inject it to achieve exact global cancellation in the subsequent iteration. However, the asynchronous activation schedule couples the effective filtering operator to which nodes transmit at each step; without the feedback term explicitly depending on the (unobserved) activation pattern of neighbors, the claimed exactness guarantee does not follow from the local information available at each node.
  2. [§3–§4] §3–§4 (closed-form derivations): the abstract and introduction assert 'rigorous theoretical analysis' yielding closed-form optimal coefficients, yet the main text provides only the final expressions without the intermediate steps that isolate the quantization noise term and solve the resulting quadratic optimization. This omission prevents verification that the noise-reduction claim remains exact rather than approximate once the three scenario-specific constraints are imposed.
minor comments (2)
  1. [§2] The quantization noise model is introduced as additive and independent, but the paper does not state whether the quantizer is uniform, dithered, or mid-rise; this detail affects the validity of the error-feedback optimality equations.
  2. [Figure 3] Figure 3 caption and surrounding text use 'error floor' without defining the metric (e.g., steady-state MSE or relative error); consistency with the theoretical expressions in Eq. (12) would improve clarity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address each major comment below and indicate the corresponding revisions to the manuscript.

read point-by-point responses

  1. Referee: [§4.3] §4.3 (random node-asynchronous updates): the derivation of optimal feedback coefficients assumes each node can locally isolate the precise quantization error e = x − Q(x) and inject it to achieve exact global cancellation in the subsequent iteration. However, the asynchronous activation schedule couples the effective filtering operator to which nodes transmit at each step; without the feedback term explicitly depending on the (unobserved) activation pattern of neighbors, the claimed exactness guarantee does not follow from the local information available at each node.

    Authors: We thank the referee for this observation. In Section 4.3 the asynchronous updates are modeled via independent Bernoulli activation variables whose statistics are known locally. The optimal feedback coefficients are derived by minimizing the expected quantization-noise variance in the subsequent global state, where the expectation is taken over the random activation pattern. Each node computes its local quantization error from the value it transmits and applies the (scalar) feedback coefficient to that error before transmission. Substituting the feedback term into the update equation shows that the expected noise contribution vanishes exactly, without any node needing to observe the instantaneous activation pattern of its neighbors. This is the standard notion of exact compensation for random asynchronous algorithms. We will add a clarifying paragraph in the revised Section 4.3 stating that the guarantee holds in expectation over the activation schedule and briefly contrast it with a path-wise guarantee. revision: partial

  2. Referee: [§3–§4] §3–§4 (closed-form derivations): the abstract and introduction assert 'rigorous theoretical analysis' yielding closed-form optimal coefficients, yet the main text provides only the final expressions without the intermediate steps that isolate the quantization noise term and solve the resulting quadratic optimization. This omission prevents verification that the noise-reduction claim remains exact rather than approximate once the three scenario-specific constraints are imposed.

    Authors: We agree that the intermediate algebraic steps were omitted for conciseness. In the revised manuscript we will expand Sections 3 and 4 to include the full derivation for each scenario: (i) write the quantized recursion, (ii) substitute the linear error-feedback term, (iii) collect the effective noise term, (iv) form the quadratic objective in the feedback coefficients, and (v) solve the resulting linear system to obtain the closed-form expressions. These steps will make explicit that the compensation is exact (the noise term is driven to zero) once the scenario-specific constraints are imposed. revision: yes

Circularity Check

0 steps flagged

No circularity: closed-form derivations are independent of results

full rationale

The paper adapts error spectrum shaping from state-space digital filters to graph filtering and derives closed-form optimal error feedback coefficients for three scenarios using standard quantization noise models and graph signal processing. These derivations rely on explicit update rules and noise isolation assumptions that are stated upfront rather than fitted or defined in terms of the final performance claims. No self-citation chains, ansatzes smuggled via prior work, or predictions that reduce to input fits by construction are present. The analysis remains self-contained against external benchmarks in digital filtering and graph theory.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The framework rests on adapting error spectrum shaping assumptions to graph domains with no explicit free parameters or new entities stated; relies on standard modeling of quantization as additive noise that can be exactly tracked.

axioms (1)
  • domain assumption Quantization noise can be isolated and quantitatively fed back for exact compensation in the graph filtering update process.
    This is the core premise enabling the framework, drawn from state-space digital filter techniques and applied to the three graph scenarios.

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