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Hilbert Transform on Graphs: Let There Be Phase

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Pith's one-line read The minimal edge additions that make a directed graph's adjacency diagonalizable and invertible always create a cycle cover, and this cycle cover is what lets a graph Hilbert transform give every node a phase.

desk verdict Useful graph-Hilbert-transform method with a fixable proof gap in its cycle-cover proposition; worth engaging after a revision. read the letter →

arxiv 2412.18501 v3 pith:FFICCWLS submitted 2024-12-24 eess.SP

Chun Hei Michael Chan , Alexandre Cionca , Dimitri Van De Ville This is my paper · ORCID
classification eess.SP MSC 05C5005C2094A12
keywords graphsignalprocessingHilberttransformdirectedgraphscyclecoverFourierinstantaneousphaseanalyticalminimaledgeperturbation
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper tackles a core obstacle in signal processing on directed graphs: the adjacency matrix is generally not diagonalizable, so the usual graph Fourier transform, and any phase derived from it, is not defined. It takes a recently proposed procedure that adds a minimal number of edges to make the adjacency matrix diagonalizable and invertible, and proves that those added edges always create a cycle cover, a set of directed cycles that visits every node. On this cycle cover the paper defines a graph Hilbert transform whose spectral filter rotates the coefficients of conjugate eigenvalue pairs by plus or minus 90 degrees, recovering the classical notions of instantaneous amplitude and phase on each cycle. A sympathetic reader would take the contribution to be: with a few well-chosen added edges, phase analysis becomes possible on any directed graph, not just on graphs whose adjacency happens to be diagonalizable.

What carries the argument

The central object is the spectral filter $\hat{H}$ defined in Eq. (1), a diagonal matrix in the eigenbasis of the perturbed adjacency $A' = U\Lambda U^{-1}$. It assigns $-j$ to eigenvalues with positive imaginary part, $+j$ to those with negative imaginary part, and $0$ to real eigenvalues, so the transform $H(x) = U\hat{H}U^{-1}x$ maps real graph signals to real graph signals and defines the analytic graph signal $\tilde{x} = x + jH(x)$. The enabling graph-theoretic fact is Proposition 1: because the perturbed graph is diagonalizable and invertible, it cannot be $r$-acyclic for nonzero $r$, so it is $0$-acyclic and therefore admits a cycle cover; each cycle in that cover behaves like a discrete periodic signal and supplies the local phase reference that makes the Hilbert interpretation meaningful.

What would settle it

Run the edge-addition procedure on a small directed graph that has a node of zero in-degree or zero out-degree, and inspect the resulting graph: if that node is left off every directed cycle, Proposition 1 is false and the phase interpretation fails there. A concrete candidate is a directed star or a directed path with a single added edge; the claimed cycle cover must include all nodes.

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Extended reading notes

Core claim

The paper's central claim is that the minimal perturbation that makes a directed graph's adjacency matrix diagonalizable and invertible also gives the graph a cycle cover, and that this cycle cover is necessary for the Hilbert transform to deliver phase information across the whole graph. The proposed graph Hilbert transform acts in the spectral domain: coefficients belonging to eigenvalues with positive imaginary part are multiplied by $-j$, those with negative imaginary part by $+j$, and real-eigenvalue coefficients are left alone, so the analytic graph signal $\tilde{x} = x + jH(x)$ yields an instantaneous amplitude and phase at every node. On a single directed cycle the construction reduces to the traditional Hilbert transform, and on graphs whose subcycles overlap, the amplitude and phase of the combined signal follow explicit combination rules derived from the contributing cycles. The paper demonstrates the contrast on a synthetic graph with fan cycles, where a Jordan-normal-form Hilbert transform cancels the signal on the fans while the proposed transform produces the expected phase shift.

Load-bearing premise

The proof of Proposition 1 depends on the cited theorem that a graph is $r$-acyclic exactly when every subgraph adjacency matrix has at least $r$ zero eigenvalues, together with the interpretation that $0$-acyclic means a cycle cover exists; if that theorem uses a different definition of $r$-acyclic, the inference from invertibility to a cycle cover does not go through.

Editorial extensions

If this is right

  • After the minimal edge perturbation, every node lies on at least one directed cycle, so the Hilbert transform yields an instantaneous amplitude and phase at every node, not just on the diagonalizable part of the graph.
  • On a single directed cycle the proposed transform is exactly the classical Hilbert transform, so standard intuitions about envelopes and phase shifts carry over.
  • For graphs whose subcycles share nodes, Corollary 1.1 gives explicit formulas for combining per-cycle amplitudes and phases into the full graph signal's amplitude and phase.
  • Because instantaneous phase can be unwrapped along cycles, amplitude and frequency modulation analysis becomes available on directed graphs.
  • Graphs whose adjacency is already diagonalizable and invertible, such as the regular 2D grid in the experiments, need no added edges and the transform produces the expected $\pi/2$ phase shift along the wave propagation direction.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Editorial inference: if Proposition 1 holds for every directed graph, the diagonalizability obstruction to phase analysis is removed in full generality, so any directed graph can be phase-analyzed by accepting the added edges and treating the resulting cycles as the periodic structure of the signal.
  • Editorial inference: Corollary 1.1 implies interference at nodes shared by several cycles; a natural test the paper does not run is to place two subcycle signals with different frequencies on overlapping cycles and look for amplitude beats at their intersection.
  • Editorial inference: since the filter only requires conjugate eigenvalue pairs, the same $\pm j$ construction could be transferred to other graph shift operators with complex spectra, such as a directed Laplacian or polar-decomposition-based operators, which the paper lists only as future directions.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. The manuscript proposes a graph Hilbert transform (GHT) for directed graphs. The authors build on the Seifert–Püschel framework that adds edges to a digraph's adjacency matrix to make it diagonalizable and invertible, define a spectral filter in Eq. (1) that multiplies GFT coefficients by −j or +j according to the sign of the imaginary part of the corresponding eigenvalue, and define an analytic graph signal. The main theoretical contribution is Proposition 1, which claims that the perturbed graph admits a cycle cover, and Corollary 1.1, which gives amplitude/phase combination rules for signals on subcycles. Experiments on a synthetic "rosace" graph and on Manhattan/regular-grid graphs illustrate that the GHT produces π/2 phase shifts along directed cycles and interpretable instantaneous amplitude/frequency.

Significance. If the claims are properly supported, the paper makes a useful contribution to graph signal processing by extending a meaningful phase notion to directed graphs with non-diagonalizable adjacency. The filter definition is simple, has no free parameters, and is shown to reduce to the standard Hilbert transform on a single directed cycle; the public implementation is a strength. The main caveat is that the paper's central structural result (Proposition 1) is not proved as printed, and the proof of Corollary 1.1 is missing from the arXiv version; both need to be repaired before the theoretical claims can be assessed. The experimental section is illustrative rather than a systematic benchmark.

major comments (2)
  1. [III-C, Proposition 1] The proof as written is not valid. The manuscript defines a graph G to be r-acyclic if any collection of vertex-disjoint cycles covers at most N−r nodes. Under this definition every graph is 0-acyclic (any collection covers at most N nodes), so the statement "therefore G′ can only be 0-acyclic, consequently G′ admits a cycle cover" does not follow: the contradiction only shows that G′ is not r-acyclic for any r>0. The inference to a cycle cover requires the definition of r-acyclic used in [21, Th. 4.4], which appears to be a deficiency-type condition, not the definition printed here. Please either state the correct definition from [21] or replace the argument with a direct determinant proof: since A′ is invertible, det(A′)≠0, so the determinant expansion contains a permutation π with A′_{i,π(i)}≠0 for all i; the cycles of π then form a vertex-disjoint cycle cover of V. This is a local but load-bearing fix, because Proposition 1 is the basis for interpreting the GHT over a cycle cover.
  2. [III-C, Corollary 1.1] The proof is deferred to a Supplementary Material file that is not present in the arXiv version (v3). As submitted, the result is unverifiable. Since this corollary is used to justify the amplitude/phase interpretation on overlapping subcycles, either include the proof in the paper or make the supplementary material available with the submission.
minor comments (5)
  1. [IV-A.1] The sentence "for a the total number of nodes of NCNF" appears garbled; if each of the NC central nodes has an outgoing fan of NF nodes, the total number of nodes is NC(NF+1), not NCNF. Please clarify.
  2. [IV-B.1] "Removing the non-trivial Jordan blocks" is inaccurate as a description of the algorithm, which adds edges to dismantle Jordan blocks; please rephrase.
  3. [Eq. (3)] The instantaneous frequency definition uses "k+1 indicates the next node on the fan"; this only makes sense on cycles with an explicit ordering, so please state this restriction when defining ω(x)[k].
  4. [Throughout] Typos: "diagonizable" in Section IV-B.1, "Theorem. 1" in the proof of Proposition 1, and the "S M" notation in Proposition 1 should be cleaned up.
  5. [Fig. 2] The claim that average amplitude and frequency per fan are "accurate estimates of the ground truth" is not quantified; please add at least one error measure or specify that the plot is qualitative.

Circularity Check

1 steps flagged · score 2.0 of 10

Minimal circularity: the GHT is an explicit definition whose single-cycle equivalence holds by construction; the cycle-cover theorem rests on an external result rather than on the paper's own claims.

  1. self definitional [Section III-B, Eq. (1); Section III-C first paragraph; Abstract]
    "We introduce the GHT by defining the following filter in the spectral domain by the diagonal matrix \hat{H}: \hat{H}[k,k] = -j if imag(\lambda_k) > 0, +j if imag(\lambda_k) < 0, 0 if imag(\lambda_k) = 0. (1) In traditional signal processing, the Hilbert transform of a signal provides the magnitude of its envelope and phase of its oscillatory pattern."

    Equation (1) is exactly the classical Hilbert transfer function -j sgn(imag(\lambda)) applied eigenvalue-wise. On a directed cycle, the adjacency eigenvalues are the DFT roots of unity, so the sign of imag(\lambda_k) coincides with the sign of the discrete frequency; therefore the paper's statement that the proposed GHT is 'equivalent to the generalized Hilbert Transform on a single cycle' is true by construction of the filter, not by an independent derivation. This is a mild self-definitional consistency, not a fitted-parameter or self-citation problem, because the paper explicitly says 'we introduce the GHT by defining' and the cycle-cover claim does not depend on this definition.

full rationale

The main theoretical step, Proposition 1, is not circular: it derives the existence of a cycle cover from invertibility of the perturbed adjacency via the external theorem [21, Th. 4.4]. The GHT itself is introduced as a definition rather than fitted to data, and the experiments use hand-constructed ground-truth signals to verify that the definition recovers the expected phase and amplitude behavior; no parameter is fitted and then renamed a prediction. The only mild construction-dependent point is that matching the classical Hilbert transform on a single directed cycle is built into the filter. The only self-citation, [25] in the discussion of monogenic signals, is not load-bearing. Two non-circular correctness and verifiability concerns should be noted separately: (i) the proof of Proposition 1 as printed appears invalid because '0-acyclic' under the paper's stated definition ('any collection of vertex-disjoint cycles covers at most N nodes') is trivially true for every graph, so the inference from the contradiction to the existence of a cycle cover requires [21]'s exact definition or a direct determinant argument; and (ii) the proof of Corollary 1.1 is deferred to supplementary material not included in the arXiv version. These issues affect rigorous verifiability but do not make the derivation circular.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No free parameters are fitted. The main axioms are the cited theorem on r-acyclic graphs and the prior perturbation framework. The GHT filter definition is the paper's central design assumption.

assumptions (4)
  • standard math Theorem 1 from [21]: a graph is r-acyclic iff every subgraph adjacency matrix in S_G has at least r zero eigenvalues; used in the proof of Proposition 1.
    The paper relies on the exact statement of this theorem to infer that a non-invertible adjacency (zero eigenvalue) implies the graph is not r-acyclic for r>0. The correctness of the proof depends on the precise definition of r-acyclic in [21].
  • domain assumption The perturbation framework of Seifert and Puschel [14] guarantees a minimal edge addition yielding an adjacency matrix A' that is both diagonalizable and invertible.
    The entire construction is built on the prior result [14]; the paper does not reprove it and uses it as a black box. If [14]'s guarantee is not exactly as stated, the cycle cover result would not apply.
  • standard math A real matrix has eigenvalues that are real or in complex conjugate pairs with corresponding conjugate eigenvectors (Horn and Johnson [17]).
    This underpins the claim that the GFT of a real graph signal has conjugate coefficients, which makes the GHT output real.
  • ad hoc to paper The graph Hilbert transform is defined by the spectral filter in Eq. (1), assigning -j, +j, or 0 based on the sign of the imaginary part of the adjacency eigenvalue.
    This is a design choice proposed by the authors, not derived from prior results. It is the core definitional step that connects complex eigenvalues to phase.

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Pith. "Pith review of Hilbert Transform on Graphs: Let There Be Phase." pith.science (2026). https://pith.science/paper/FFICCWLS

@misc{pith2026241218501,
  author       = {Pith},
  title        = {Pith review of: Hilbert Transform on Graphs: Let There Be Phase},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FFICCWLS}},
  note         = {Machine review of arXiv:2412.18501}
}
read the original abstract

In the past years, many signal processing operations have been successfully adapted to the graph setting. One elegant and effective approach is to exploit the eigendecomposition of a graph shift operator (GSO), such as the adjacency or Laplacian operator, to define a graph Fourier transform when projecting graph signals on the corresponding basis. However, the extension of this scheme to directed graphs is challenging since the associated GSO is non-symmetric and, in general, not diagonalizable. Here, we build upon a recent framework that adds a minimal number of edges to allow diagonalization of the GSO and thus provide a proper graph Fourier transform. Furthermore, we show that such minimal addition of edges creates a cycle cover and that it is essential for the phase analysis of a signal throughout the graph. Concurrently, we propose a generalization of the Hilbert transform interpreted over the newfound cycle cover, which re-establishes intuitions from traditional Hilbert Transform, equivalent to the generalized Hilbert Transform on a single cycle. This generalization leads to a number of simple and elegant recipes to effectively exploit the phase information of graph signals provided by the graph Fourier transform. The feasibility of the approach is demonstrated on several examples.

Figures

Figures reproduced from arXiv: 2412.18501 by the authors.

Figure 2
Figure 2. (a) Instantaneous amplitude distribution across fans [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. (a) Manhattan graph. The green edges indicate those that are added by the perturbation. (b) Regular 2D grid with periodic boundary conditions along [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

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