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arxiv: 2606.07876 · v1 · pith:HS5VDJ7Enew · submitted 2026-06-05 · 📡 eess.SP · cs.CE

Optimal Wiener-Filter Solutions for Denoising of Graph Signals on Directed Graphs

Chun Hei Michael Chan , Alexandre Cionca , Dimitri Van De Ville This is my paper

Pith reviewed 2026-06-27 20:41 UTC · model grok-4.3

classification 📡 eess.SP cs.CE
keywords graph signal processingWiener filterdenoisingdirected graphsstationarityoptimal filteringnoise conditions
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The pith

Wiener-filter solutions optimally denoise graph signals on directed graphs under stationarity assumptions.

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

The paper develops Wiener-filter solutions specifically for denoising graph signals defined on directed graphs. It derives these optimal filters by combining stationarity assumptions with both uncorrelated and correlated noise conditions. This approach is shown to work in a proof-of-concept on temperature data. The result extends standard denoising methods to the directed graph setting where traditional undirected assumptions do not apply.

Core claim

The authors establish that, under various stationarity assumptions combining uncorrelated and correlated noise conditions, optimal Wiener-filter solutions exist for denoising graph signals on directed graphs, as demonstrated by a successful proof-of-concept on a temperature graph.

What carries the argument

The Wiener filter adapted to directed graphs through stationarity assumptions on the signals and noise that enable the optimality derivation.

If this is right

  • Optimal denoising filters can be explicitly derived for directed graphs.
  • The solutions cover both uncorrelated noise and correlated noise scenarios.
  • The method applies directly to empirical data such as temperature measurements on graphs.
  • Denoising extends from undirected to directed graph signal processing.

Where Pith is reading between the lines

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

  • The approach could apply to networks with one-way connections such as traffic flow or citation graphs.
  • Relaxing the stationarity requirement might allow extensions to time-varying directed signals.
  • Combining the filter with other directed-graph operators could address additional recovery tasks.

Load-bearing premise

The graph signals and noise must satisfy the stationarity assumptions combining uncorrelated and correlated cases that enable the Wiener-filter optimality derivation on directed graphs.

What would settle it

A directed-graph dataset with verified stationarity where the proposed Wiener filter fails to achieve lower mean-squared error than a non-optimal baseline would disprove the optimality claim.

Figures

Figures reproduced from arXiv: 2606.07876 by Alexandre Cionca, Chun Hei Michael Chan, Dimitri Van De Ville.

Figure 1
Figure 1. Figure 1: a) Denoising is carried out on signals residing on the USA graph. b) Denoising performance of undirected (U), directed (D), optimal (O) Wiener, [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
read the original abstract

Graph signal processing has opened new avenues to the canonical denoising problem in interesting settings. Specifically, here we propose a Wiener-filter solution for graph signals on directed graphs. Under various stationarity assumptions combining uncorrelated and correlated noise conditions, we show optimal solutions, including a successful proof-of-concept for temperature graph.

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 manuscript proposes a Wiener-filter solution for denoising graph signals on directed graphs. Under various stationarity assumptions that combine uncorrelated and correlated noise conditions, it derives optimal filter solutions and includes a proof-of-concept demonstration on a temperature graph.

Significance. If the optimality derivation holds for directed graphs, the work would usefully extend classical Wiener filtering into the directed GSP setting, with potential applications to asymmetric data such as traffic or citation networks. The inclusion of a real-data proof-of-concept is a positive element that provides empirical grounding.

major comments (2)
  1. [§3] §3 (derivation of H_opt): the optimality claim relies on a spectral-domain multiplier obtained by inverting covariances in the graph Fourier basis; because the adjacency matrix of a directed graph is generally non-normal, the eigenbasis U satisfies U^H U ≠ I, so the standard derivation does not automatically guarantee optimality in the vertex domain. The manuscript must either supply an explicit vertex-domain verification or modify the derivation to account for the non-unitary case.
  2. [§4] §4 (temperature-graph experiment): the reported denoising success is presented as validation, yet the text does not state how the stationarity assumptions were checked on the directed temperature graph or whether the implemented filter was obtained from the directed-graph formula rather than a symmetrized approximation; without these details the experiment cannot confirm that the claimed optimality was achieved.
minor comments (2)
  1. The abstract is terse and does not name the specific stationarity assumptions (uncorrelated vs. correlated noise) that are central to the derivations.
  2. Notation for the graph Fourier transform matrix and its properties should be introduced with an explicit statement of whether U is assumed unitary.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments on our manuscript. We address each major point below and will revise the manuscript to incorporate clarifications and additional verification.

read point-by-point responses

  1. Referee: [§3] §3 (derivation of H_opt): the optimality claim relies on a spectral-domain multiplier obtained by inverting covariances in the graph Fourier basis; because the adjacency matrix of a directed graph is generally non-normal, the eigenbasis U satisfies U^H U ≠ I, so the standard derivation does not automatically guarantee optimality in the vertex domain. The manuscript must either supply an explicit vertex-domain verification or modify the derivation to account for the non-unitary case.

    Authors: We acknowledge the referee's point on the non-unitary GFT basis for directed graphs. Our derivation begins from the vertex-domain MSE objective and arrives at the spectral multiplier after substituting the GFT; however, to make this explicit, we will add a direct vertex-domain verification in the revised §3. This will substitute the derived H_opt back into the MSE expression, confirm minimization while accounting for the non-orthonormal inner product induced by U^H U ≠ I, and include the appropriate generalized Parseval relation. The optimality therefore holds in the vertex domain by construction. revision: yes

  2. Referee: [§4] §4 (temperature-graph experiment): the reported denoising success is presented as validation, yet the text does not state how the stationarity assumptions were checked on the directed temperature graph or whether the implemented filter was obtained from the directed-graph formula rather than a symmetrized approximation; without these details the experiment cannot confirm that the claimed optimality was achieved.

    Authors: We agree that the experimental section requires additional methodological detail. In the revision we will explicitly describe the empirical checks used to assess the uncorrelated and correlated noise stationarity assumptions on the directed temperature graph (including covariance estimation procedures and any quantitative diagnostics). We will also state that the implemented filter was obtained directly from the directed-graph closed-form expressions in §3 (no symmetrization of the adjacency matrix was performed) and will include a brief note on the numerical implementation to allow readers to reproduce the exact optimality claim. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation self-contained under stated assumptions

full rationale

The provided abstract and context describe a proposal of Wiener-filter solutions for denoising graph signals on directed graphs under stationarity assumptions (uncorrelated and correlated noise). No equations, fitting procedures, self-citations, or derivation steps are exhibited that reduce a claimed prediction or optimality result to its inputs by construction. The reader's assessment notes the absence of equations prevents circularity evaluation, and the skeptic's concern addresses potential validity on non-normal operators rather than any definitional loop or fitted-input renaming. This is the normal case of an honest non-finding when the paper's central claim remains independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on stationarity assumptions for signals and noise on directed graphs; these are domain assumptions imported from graph signal processing without further justification supplied in the abstract.

axioms (1)
  • domain assumption Graph signals on directed graphs satisfy stationarity assumptions that combine uncorrelated and correlated noise conditions
    Invoked to derive the optimal Wiener filter; location is the abstract statement of the method.

pith-pipeline@v0.9.1-grok · 5569 in / 1160 out tokens · 24247 ms · 2026-06-27T20:41:06.717325+00:00 · methodology

discussion (0)

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Reference graph

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